Logiweb expansion of sup2 in pyk

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"File page.tex
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\floatplacement{figure}{h!}
\floatplacement{table}{h!}
\hyphenation{her-ud-over ek-si-stens-va-ri-ab-le an-dre dob-belt-im-pli-ka-ti-on ob-jekt-kvan-tor de-fi-ni-tions-lem-ma ens-be-tyd-en-de
und-er-af-snit slut-ning-en inde-hol-der for-klar-ing-en si-de-be-ting-el-sen
des-uden be-vis-check-er-en}

" [ ragged right ] "

SUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUP

" [ math define statement of prop lemma to negated double imply as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar infer metavar var b end metavar infer not0 metavar var c end metavar infer not0 metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar end define end math ] "

" [ math define proof of prop lemma to negated double imply as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar infer metavar var b end metavar infer not0 metavar var c end metavar infer metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar infer prop lemma mp2 modus ponens metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar modus ponens metavar var a end metavar modus ponens metavar var b end metavar conclude metavar var c end metavar cut prop lemma from contradiction modus ponens metavar var c end metavar modus ponens not0 metavar var c end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar infer metavar var b end metavar infer not0 metavar var c end metavar infer metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar infer not0 metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar conclude metavar var a end metavar imply metavar var b end metavar imply not0 metavar var c end metavar imply metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply not0 metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar cut metavar var a end metavar infer metavar var b end metavar infer not0 metavar var c end metavar infer prop lemma mp3 modus ponens metavar var a end metavar imply metavar var b end metavar imply not0 metavar var c end metavar imply metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply not0 metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar modus ponens metavar var a end metavar modus ponens metavar var b end metavar modus ponens not0 metavar var c end metavar conclude metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply not0 metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar cut prop lemma imply negation modus ponens metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply not0 metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of prop lemma from negated and (imply) as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 not0 metavar var a end metavar imply not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var b end metavar end define end math ] "

" [ math define proof of prop lemma from negated and (imply) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 not0 metavar var a end metavar imply not0 metavar var b end metavar infer metavar var a end metavar infer prop lemma from negated and modus ponens not0 not0 metavar var a end metavar imply not0 metavar var b end metavar modus ponens metavar var a end metavar conclude not0 metavar var b end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed not0 not0 metavar var a end metavar imply not0 metavar var b end metavar infer metavar var a end metavar infer not0 metavar var b end metavar conclude not0 not0 metavar var a end metavar imply not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of prop lemma remove double neg (consequent) as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply not0 not0 metavar var b end metavar infer metavar var a end metavar imply metavar var b end metavar end define end math ] "

" [ math define proof of prop lemma remove double neg (consequent) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply not0 not0 metavar var b end metavar infer metavar var a end metavar infer 1rule mp modus ponens metavar var a end metavar imply not0 not0 metavar var b end metavar modus ponens metavar var a end metavar conclude not0 not0 metavar var b end metavar cut prop lemma remove double neg modus ponens not0 not0 metavar var b end metavar conclude metavar var b end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply not0 not0 metavar var b end metavar infer metavar var a end metavar infer metavar var b end metavar conclude metavar var a end metavar imply not0 not0 metavar var b end metavar imply metavar var a end metavar imply metavar var b end metavar cut metavar var a end metavar imply not0 not0 metavar var b end metavar infer 1rule mp modus ponens metavar var a end metavar imply not0 not0 metavar var b end metavar imply metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar imply not0 not0 metavar var b end metavar conclude metavar var a end metavar imply metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of pred lemma (A)to(~E~)(Imply) as system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed for all objects metavar var v1 end metavar indeed metavar var a end metavar imply not0 not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar end define end math ] "

" [ math define proof of pred lemma (A)to(~E~)(Imply) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed for all objects metavar var v1 end metavar indeed metavar var a end metavar infer lemma a4 at metavar var v1 end metavar modus ponens for all objects metavar var v1 end metavar indeed metavar var a end metavar conclude metavar var a end metavar cut prop lemma add double neg modus ponens metavar var a end metavar conclude not0 not0 metavar var a end metavar cut 1rule gen modus ponens not0 not0 metavar var a end metavar conclude for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar cut prop lemma add double neg modus ponens for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar conclude not0 not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar cut 1rule repetition modus ponens not0 not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar conclude not0 not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar cut all metavar var v1 end metavar indeed all metavar var a end metavar indeed 1rule deduction modus ponens all metavar var v1 end metavar indeed all metavar var a end metavar indeed for all objects metavar var v1 end metavar indeed metavar var a end metavar infer not0 not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar conclude for all objects metavar var v1 end metavar indeed metavar var a end metavar imply not0 not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of pred lemma (E)to(~A~)(Imply) as system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar end define end math ] "

" [ math define proof of pred lemma (E)to(~A~)(Imply) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed prop lemma auto imply conclude not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar cut 1rule repetition modus ponens not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar conclude not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of pred lemma (E~)to(~A)(Imply) as system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed metavar var a end metavar end define end math ] "

" [ math define proof of pred lemma (E~)to(~A)(Imply) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar infer prop lemma add double neg modus ponens not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar conclude not0 not0 not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar cut pred lemma (A)to(~E~)(Imply) conclude for all objects metavar var v1 end metavar indeed metavar var a end metavar imply not0 not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar cut prop lemma mt modus ponens for all objects metavar var v1 end metavar indeed metavar var a end metavar imply not0 not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar modus ponens not0 not0 not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar conclude not0 for all objects metavar var v1 end metavar indeed metavar var a end metavar cut all metavar var v1 end metavar indeed all metavar var a end metavar indeed 1rule deduction modus ponens all metavar var v1 end metavar indeed all metavar var a end metavar indeed not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar infer not0 for all objects metavar var v1 end metavar indeed metavar var a end metavar conclude not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of pred lemma addNegatedAll as system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var b end metavar imply metavar var a end metavar infer not0 for all objects metavar var v1 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed metavar var b end metavar end define end math ] "

" [ math define proof of pred lemma addNegatedAll as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var b end metavar imply metavar var a end metavar infer pred lemma addAll modus ponens metavar var b end metavar imply metavar var a end metavar conclude for all objects metavar var v1 end metavar indeed metavar var b end metavar imply for all objects metavar var v1 end metavar indeed metavar var a end metavar cut prop lemma contrapositive modus ponens for all objects metavar var v1 end metavar indeed metavar var b end metavar imply for all objects metavar var v1 end metavar indeed metavar var a end metavar conclude not0 for all objects metavar var v1 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of pred lemma toNegatedAEA as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var v3 end metavar indeed all metavar var a end metavar indeed not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar end define end math ] "

" [ math define proof of pred lemma toNegatedAEA as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var v3 end metavar indeed all metavar var a end metavar indeed not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar infer pred lemma (E~)to(~A)(Imply) conclude not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar imply not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar cut pred lemma addNegatedAll modus ponens not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar imply not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar conclude not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar cut pred lemma (E)to(~A~)(Imply) conclude not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar cut prop lemma imply transitivity modus ponens not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar modus ponens not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar conclude not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar cut pred lemma addNegatedAll modus ponens not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar conclude not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar cut pred lemma (E)to(~A~)(Imply) conclude not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar cut prop lemma imply transitivity modus ponens not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar modus ponens not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar conclude not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar cut 1rule mp modus ponens not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar modus ponens not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar conclude not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of pred lemma (A~)to(~E) as system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar infer not0 not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar end define end math ] "

" [ math define proof of pred lemma (A~)to(~E) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar infer prop lemma add double neg modus ponens for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar conclude not0 not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar cut 1rule repetition modus ponens not0 not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar conclude not0 not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of pred lemma exist mp3 as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var v3 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar infer not0 for all objects metavar var v2 end metavar indeed not0 metavar var b end metavar infer not0 for all objects metavar var v3 end metavar indeed not0 metavar var c end metavar infer metavar var d end metavar end define end math ] "

" [ math define proof of pred lemma exist mp3 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var v3 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar infer not0 for all objects metavar var v2 end metavar indeed not0 metavar var b end metavar infer not0 for all objects metavar var v3 end metavar indeed not0 metavar var c end metavar infer pred lemma exist mp2 modus ponens metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar modus ponens not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar modus ponens not0 for all objects metavar var v2 end metavar indeed not0 metavar var b end metavar conclude metavar var c end metavar imply metavar var d end metavar cut pred lemma exist mp modus ponens metavar var c end metavar imply metavar var d end metavar modus ponens not0 for all objects metavar var v3 end metavar indeed not0 metavar var c end metavar conclude metavar var d end metavar end quote state proof state cache var c end expand end define end math ] "





" [ math define statement of lemma positiveToLeft(Eq) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar + metavar var z end metavar infer metavar var x end metavar + - metavar var z end metavar = metavar var y end metavar end define end math ] "

" [ math define proof of lemma positiveToLeft(Eq) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar + metavar var z end metavar infer lemma eqAddition modus ponens metavar var x end metavar = metavar var y end metavar + metavar var z end metavar conclude metavar var x end metavar + - metavar var z end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma x=x+y-y conclude metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar conclude metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar + - metavar var z end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar modus ponens metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar conclude metavar var x end metavar + - metavar var z end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "




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" [ math define statement of lemma expZero exact as system Q infer all metavar var x end metavar indeed metavar var x end metavar ^ 0 = 1 end define end math ] "

" [ math define proof of lemma expZero exact as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed lemma eqReflexivity conclude 0 = 0 cut 1rule expZero modus ponens 0 = 0 conclude metavar var x end metavar ^ 0 = 1 end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma +1IsPositive(N) as system Q infer all metavar var m end metavar indeed Nat( metavar var m end metavar ) endorse not0 0 <= metavar var m end metavar + 1 imply not0 not0 0 = metavar var m end metavar + 1 end define end math ] "

" [ math define proof of lemma +1IsPositive(N) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed Nat( metavar var m end metavar ) endorse axiom nonnegative(N) modus probans Nat( metavar var m end metavar ) conclude 0 <= metavar var m end metavar cut lemma leqPlus1 modus ponens 0 <= metavar var m end metavar conclude not0 0 <= metavar var m end metavar + 1 imply not0 not0 0 = metavar var m end metavar + 1 end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma sameExp base as system Q infer all metavar var n end metavar indeed all metavar var x end metavar indeed for all objects metavar var n end metavar indeed 0 = metavar var n end metavar imply metavar var x end metavar ^ 0 = metavar var x end metavar ^ metavar var n end metavar end define end math ] "

" [ math define proof of lemma sameExp base as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var n end metavar indeed all metavar var x end metavar indeed 0 = metavar var n end metavar infer lemma expZero exact conclude metavar var x end metavar ^ 0 = 1 cut lemma eqSymmetry modus ponens 0 = metavar var n end metavar conclude metavar var n end metavar = 0 cut 1rule expZero modus ponens metavar var n end metavar = 0 conclude metavar var x end metavar ^ metavar var n end metavar = 1 cut lemma eqSymmetry modus ponens metavar var x end metavar ^ metavar var n end metavar = 1 conclude 1 = metavar var x end metavar ^ metavar var n end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar ^ 0 = 1 modus ponens 1 = metavar var x end metavar ^ metavar var n end metavar conclude metavar var x end metavar ^ 0 = metavar var x end metavar ^ metavar var n end metavar cut all metavar var n end metavar indeed all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var n end metavar indeed all metavar var x end metavar indeed 0 = metavar var n end metavar infer metavar var x end metavar ^ 0 = metavar var x end metavar ^ metavar var n end metavar conclude 0 = metavar var n end metavar imply metavar var x end metavar ^ 0 = metavar var x end metavar ^ metavar var n end metavar cut 1rule gen modus ponens 0 = metavar var n end metavar imply metavar var x end metavar ^ 0 = metavar var x end metavar ^ metavar var n end metavar conclude for all objects metavar var n end metavar indeed 0 = metavar var n end metavar imply metavar var x end metavar ^ 0 = metavar var x end metavar ^ metavar var n end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma sameExp indu as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var x end metavar indeed for all objects metavar var n end metavar indeed metavar var m end metavar = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar imply for all objects metavar var n end metavar indeed metavar var m end metavar + 1 = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar ^ metavar var n end metavar end define end math ] "

" [ math define proof of lemma sameExp indu as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var x end metavar indeed all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var x end metavar indeed for all objects metavar var n end metavar indeed metavar var m end metavar = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar infer metavar var m end metavar + 1 = metavar var n end metavar infer lemma +1IsPositive(N) conclude not0 0 <= metavar var m end metavar + 1 imply not0 not0 0 = metavar var m end metavar + 1 cut 1rule expPositive modus ponens not0 0 <= metavar var m end metavar + 1 imply not0 not0 0 = metavar var m end metavar + 1 conclude metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 cut lemma x=x+y-y conclude metavar var m end metavar = metavar var m end metavar + 1 + - 1 cut lemma a4 at metavar var m end metavar + 1 + - 1 modus ponens for all objects metavar var n end metavar indeed metavar var m end metavar = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar conclude metavar var m end metavar = metavar var m end metavar + 1 + - 1 imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 cut 1rule mp modus ponens metavar var m end metavar = metavar var m end metavar + 1 + - 1 imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 modus ponens metavar var m end metavar = metavar var m end metavar + 1 + - 1 conclude metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 cut lemma eqSymmetry modus ponens metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 conclude metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 = metavar var x end metavar ^ metavar var m end metavar cut lemma eqMultiplicationLeft modus ponens metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 = metavar var x end metavar ^ metavar var m end metavar conclude metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 = metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar cut lemma a4 at metavar var n end metavar + - 1 modus ponens for all objects metavar var n end metavar indeed metavar var m end metavar = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar conclude metavar var m end metavar = metavar var n end metavar + - 1 imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar + - 1 cut lemma positiveToRight(Eq) modus ponens metavar var m end metavar + 1 = metavar var n end metavar conclude metavar var m end metavar = metavar var n end metavar + - 1 cut 1rule mp modus ponens metavar var m end metavar = metavar var n end metavar + - 1 imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar + - 1 modus ponens metavar var m end metavar = metavar var n end metavar + - 1 conclude metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar + - 1 cut lemma eqMultiplicationLeft modus ponens metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar + - 1 conclude metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar * metavar var x end metavar ^ metavar var n end metavar + - 1 cut lemma subLessRight modus ponens metavar var m end metavar + 1 = metavar var n end metavar modus ponens not0 0 <= metavar var m end metavar + 1 imply not0 not0 0 = metavar var m end metavar + 1 conclude not0 0 <= metavar var n end metavar imply not0 not0 0 = metavar var n end metavar cut 1rule expPositive modus ponens not0 0 <= metavar var n end metavar imply not0 not0 0 = metavar var n end metavar conclude metavar var x end metavar ^ metavar var n end metavar = metavar var x end metavar * metavar var x end metavar ^ metavar var n end metavar + - 1 cut lemma eqSymmetry modus ponens metavar var x end metavar ^ metavar var n end metavar = metavar var x end metavar * metavar var x end metavar ^ metavar var n end metavar + - 1 conclude metavar var x end metavar * metavar var x end metavar ^ metavar var n end metavar + - 1 = metavar var x end metavar ^ metavar var n end metavar cut lemma eqTransitivity5 modus ponens metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 modus ponens metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 = metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar modus ponens metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar * metavar var x end metavar ^ metavar var n end metavar + - 1 modus ponens metavar var x end metavar * metavar var x end metavar ^ metavar var n end metavar + - 1 = metavar var x end metavar ^ metavar var n end metavar conclude metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar ^ metavar var n end metavar cut 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var x end metavar indeed for all objects metavar var n end metavar indeed metavar var m end metavar = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar infer metavar var m end metavar + 1 = metavar var n end metavar infer metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar ^ metavar var n end metavar conclude for all objects metavar var n end metavar indeed metavar var m end metavar = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar imply metavar var m end metavar + 1 = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar ^ metavar var n end metavar cut for all objects metavar var n end metavar indeed metavar var m end metavar = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar infer 1rule mp modus ponens for all objects metavar var n end metavar indeed metavar var m end metavar = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar imply metavar var m end metavar + 1 = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar ^ metavar var n end metavar modus ponens for all objects metavar var n end metavar indeed metavar var m end metavar = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar conclude metavar var m end metavar + 1 = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar ^ metavar var n end metavar cut 1rule gen modus ponens metavar var m end metavar + 1 = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar ^ metavar var n end metavar conclude for all objects metavar var n end metavar indeed metavar var m end metavar + 1 = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar ^ metavar var n end metavar cut all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var x end metavar indeed for all objects metavar var n end metavar indeed metavar var m end metavar = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar infer for all objects metavar var n end metavar indeed metavar var m end metavar + 1 = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar ^ metavar var n end metavar conclude for all objects metavar var n end metavar indeed metavar var m end metavar = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar imply for all objects metavar var n end metavar indeed metavar var m end metavar + 1 = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar ^ metavar var n end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma sameExp as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var x end metavar indeed metavar var m end metavar = metavar var n end metavar infer metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar end define end math ] "

" [ math define proof of lemma sameExp as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var x end metavar indeed metavar var m end metavar = metavar var n end metavar infer lemma sameExp base conclude for all objects metavar var n end metavar indeed 0 = metavar var n end metavar imply metavar var x end metavar ^ 0 = metavar var x end metavar ^ metavar var n end metavar cut lemma sameExp indu conclude for all objects metavar var n end metavar indeed metavar var m end metavar = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar imply for all objects metavar var n end metavar indeed metavar var m end metavar + 1 = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar ^ metavar var n end metavar cut lemma induction modus ponens for all objects metavar var n end metavar indeed 0 = metavar var n end metavar imply metavar var x end metavar ^ 0 = metavar var x end metavar ^ metavar var n end metavar modus ponens for all objects metavar var n end metavar indeed metavar var m end metavar = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar imply for all objects metavar var n end metavar indeed metavar var m end metavar + 1 = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar ^ metavar var n end metavar conclude for all objects metavar var n end metavar indeed metavar var m end metavar = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar cut lemma a4 at metavar var n end metavar modus ponens for all objects metavar var n end metavar indeed metavar var m end metavar = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar conclude metavar var m end metavar = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar cut 1rule mp modus ponens metavar var m end metavar = metavar var n end metavar imply metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar modus ponens metavar var m end metavar = metavar var n end metavar conclude metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var n end metavar end quote state proof state cache var c end expand end define end math ] "

------------

" [ math define statement of lemma exp(+1) as system Q infer all metavar var m end metavar indeed all metavar var x end metavar indeed metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar end define end math ] "

" [ math define proof of lemma exp(+1) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var x end metavar indeed lemma +1IsPositive(N) conclude not0 0 <= metavar var m end metavar + 1 imply not0 not0 0 = metavar var m end metavar + 1 cut 1rule expPositive modus ponens not0 0 <= metavar var m end metavar + 1 imply not0 not0 0 = metavar var m end metavar + 1 conclude metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 cut lemma x=x+y-y conclude metavar var m end metavar = metavar var m end metavar + 1 + - 1 cut lemma sameExp modus ponens metavar var m end metavar = metavar var m end metavar + 1 + - 1 conclude metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 cut lemma eqMultiplicationLeft modus ponens metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 conclude metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 cut lemma eqSymmetry modus ponens metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 conclude metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 = metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 modus ponens metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar + 1 + - 1 = metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar conclude metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar end quote state proof state cache var c end expand end define end math ] "





" [ math define statement of lemma distributionOut(Minus) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar * metavar var y end metavar + - metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * metavar var y end metavar + - metavar var z end metavar end define end math ] "

" [ math define proof of lemma distributionOut(Minus) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed lemma times(-1)Left conclude - 1 * metavar var x end metavar * metavar var z end metavar = - metavar var x end metavar * metavar var z end metavar cut lemma eqSymmetry modus ponens - 1 * metavar var x end metavar * metavar var z end metavar = - metavar var x end metavar * metavar var z end metavar conclude - metavar var x end metavar * metavar var z end metavar = - 1 * metavar var x end metavar * metavar var z end metavar cut axiom timesCommutativity conclude - 1 * metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * metavar var z end metavar * - 1 cut axiom timesAssociativity conclude metavar var x end metavar * metavar var z end metavar * - 1 = metavar var x end metavar * metavar var z end metavar * - 1 cut lemma times(-1) conclude metavar var z end metavar * - 1 = - metavar var z end metavar cut lemma eqMultiplicationLeft modus ponens metavar var z end metavar * - 1 = - metavar var z end metavar conclude metavar var x end metavar * metavar var z end metavar * - 1 = metavar var x end metavar * - metavar var z end metavar cut lemma eqTransitivity5 modus ponens - metavar var x end metavar * metavar var z end metavar = - 1 * metavar var x end metavar * metavar var z end metavar modus ponens - 1 * metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * metavar var z end metavar * - 1 modus ponens metavar var x end metavar * metavar var z end metavar * - 1 = metavar var x end metavar * metavar var z end metavar * - 1 modus ponens metavar var x end metavar * metavar var z end metavar * - 1 = metavar var x end metavar * - metavar var z end metavar conclude - metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * - metavar var z end metavar cut lemma eqAdditionLeft modus ponens - metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * - metavar var z end metavar conclude metavar var x end metavar * metavar var y end metavar + - metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * metavar var y end metavar + metavar var x end metavar * - metavar var z end metavar cut lemma distributionOut conclude metavar var x end metavar * metavar var y end metavar + metavar var x end metavar * - metavar var z end metavar = metavar var x end metavar * metavar var y end metavar + - metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar * metavar var y end metavar + - metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * metavar var y end metavar + metavar var x end metavar * - metavar var z end metavar modus ponens metavar var x end metavar * metavar var y end metavar + metavar var x end metavar * - metavar var z end metavar = metavar var x end metavar * metavar var y end metavar + - metavar var z end metavar conclude metavar var x end metavar * metavar var y end metavar + - metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * metavar var y end metavar + - metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma (1/2)(x+y)-x=(1/2)(y-x) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 1/ 1 + 1 * metavar var x end metavar + metavar var y end metavar + - metavar var x end metavar = 1/ 1 + 1 * metavar var y end metavar + - metavar var x end metavar end define end math ] "

" [ math define proof of lemma (1/2)(x+y)-x=(1/2)(y-x) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed axiom distribution conclude 1/ 1 + 1 * metavar var x end metavar + metavar var y end metavar = 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar cut lemma eqAddition modus ponens 1/ 1 + 1 * metavar var x end metavar + metavar var y end metavar = 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar conclude 1/ 1 + 1 * metavar var x end metavar + metavar var y end metavar + - metavar var x end metavar = 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar + - metavar var x end metavar cut axiom plusCommutativity conclude 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar = 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar cut lemma eqAddition modus ponens 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar = 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar conclude 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar + - metavar var x end metavar = 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar + - metavar var x end metavar cut axiom plusAssociativity conclude 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar + - metavar var x end metavar = 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar + - metavar var x end metavar cut lemma (1/2)x+(1/2)x=x conclude 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var x end metavar = metavar var x end metavar cut lemma positiveToRight(Eq) modus ponens 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var x end metavar = metavar var x end metavar conclude 1/ 1 + 1 * metavar var x end metavar = metavar var x end metavar + - 1/ 1 + 1 * metavar var x end metavar cut lemma eqNegated modus ponens 1/ 1 + 1 * metavar var x end metavar = metavar var x end metavar + - 1/ 1 + 1 * metavar var x end metavar conclude - 1/ 1 + 1 * metavar var x end metavar = - metavar var x end metavar + - 1/ 1 + 1 * metavar var x end metavar cut lemma minusNegated conclude - metavar var x end metavar + - 1/ 1 + 1 * metavar var x end metavar = 1/ 1 + 1 * metavar var x end metavar + - metavar var x end metavar cut lemma eqTransitivity modus ponens - 1/ 1 + 1 * metavar var x end metavar = - metavar var x end metavar + - 1/ 1 + 1 * metavar var x end metavar modus ponens - metavar var x end metavar + - 1/ 1 + 1 * metavar var x end metavar = 1/ 1 + 1 * metavar var x end metavar + - metavar var x end metavar conclude - 1/ 1 + 1 * metavar var x end metavar = 1/ 1 + 1 * metavar var x end metavar + - metavar var x end metavar cut lemma eqSymmetry modus ponens - 1/ 1 + 1 * metavar var x end metavar = 1/ 1 + 1 * metavar var x end metavar + - metavar var x end metavar conclude 1/ 1 + 1 * metavar var x end metavar + - metavar var x end metavar = - 1/ 1 + 1 * metavar var x end metavar cut lemma eqAdditionLeft modus ponens 1/ 1 + 1 * metavar var x end metavar + - metavar var x end metavar = - 1/ 1 + 1 * metavar var x end metavar conclude 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar + - metavar var x end metavar = 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar cut lemma distributionOut(Minus) conclude 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar = 1/ 1 + 1 * metavar var y end metavar + - metavar var x end metavar cut lemma eqTransitivity6 modus ponens 1/ 1 + 1 * metavar var x end metavar + metavar var y end metavar + - metavar var x end metavar = 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar + - metavar var x end metavar modus ponens 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar + - metavar var x end metavar = 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar + - metavar var x end metavar modus ponens 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar + - metavar var x end metavar = 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar + - metavar var x end metavar modus ponens 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar + - metavar var x end metavar = 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar modus ponens 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar = 1/ 1 + 1 * metavar var y end metavar + - metavar var x end metavar conclude 1/ 1 + 1 * metavar var x end metavar + metavar var y end metavar + - metavar var x end metavar = 1/ 1 + 1 * metavar var y end metavar + - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma y-(1/2)(x+y)=(1/2)(y-x) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar + metavar var y end metavar = 1/ 1 + 1 * metavar var y end metavar + - metavar var x end metavar end define end math ] "

" [ math define proof of lemma y-(1/2)(x+y)=(1/2)(y-x) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed axiom distribution conclude 1/ 1 + 1 * metavar var x end metavar + metavar var y end metavar = 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar cut lemma eqNegated modus ponens 1/ 1 + 1 * metavar var x end metavar + metavar var y end metavar = 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar conclude - 1/ 1 + 1 * metavar var x end metavar + metavar var y end metavar = - 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar cut lemma eqAdditionLeft modus ponens - 1/ 1 + 1 * metavar var x end metavar + metavar var y end metavar = - 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar conclude metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar + metavar var y end metavar = metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar cut axiom plusCommutativity conclude 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar = 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar cut lemma eqNegated modus ponens 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar = 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar conclude - 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar = - 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar cut lemma -x-y=-(x+y) conclude - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar = - 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar cut lemma eqSymmetry modus ponens - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar = - 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar conclude - 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar = - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar cut lemma eqTransitivity modus ponens - 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar = - 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar modus ponens - 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var x end metavar = - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar conclude - 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar = - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar cut lemma eqAdditionLeft modus ponens - 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar = - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar conclude metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar = metavar var y end metavar + - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar cut axiom plusAssociativity conclude metavar var y end metavar + - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar = metavar var y end metavar + - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar + - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar = metavar var y end metavar + - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar conclude metavar var y end metavar + - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar = metavar var y end metavar + - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar cut lemma (1/2)x+(1/2)x=x conclude 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var y end metavar = metavar var y end metavar cut lemma positiveToRight(Eq) modus ponens 1/ 1 + 1 * metavar var y end metavar + 1/ 1 + 1 * metavar var y end metavar = metavar var y end metavar conclude 1/ 1 + 1 * metavar var y end metavar = metavar var y end metavar + - 1/ 1 + 1 * metavar var y end metavar cut lemma eqSymmetry modus ponens 1/ 1 + 1 * metavar var y end metavar = metavar var y end metavar + - 1/ 1 + 1 * metavar var y end metavar conclude metavar var y end metavar + - 1/ 1 + 1 * metavar var y end metavar = 1/ 1 + 1 * metavar var y end metavar cut lemma eqAddition modus ponens metavar var y end metavar + - 1/ 1 + 1 * metavar var y end metavar = 1/ 1 + 1 * metavar var y end metavar conclude metavar var y end metavar + - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar = 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar cut lemma distributionOut(Minus) conclude 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar = 1/ 1 + 1 * metavar var y end metavar + - metavar var x end metavar cut lemma eqTransitivity6 modus ponens metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar + metavar var y end metavar = metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar modus ponens metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var y end metavar = metavar var y end metavar + - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar modus ponens metavar var y end metavar + - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar = metavar var y end metavar + - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar modus ponens metavar var y end metavar + - 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar = 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar modus ponens 1/ 1 + 1 * metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar = 1/ 1 + 1 * metavar var y end metavar + - metavar var x end metavar conclude metavar var y end metavar + - 1/ 1 + 1 * metavar var x end metavar + metavar var y end metavar = 1/ 1 + 1 * metavar var y end metavar + - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "




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" [ math define statement of lemma positiveBase base as system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar ^ 0 imply not0 not0 0 = metavar var x end metavar ^ 0 end define end math ] "

" [ math define proof of lemma positiveBase base as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed lemma expZero exact conclude metavar var x end metavar ^ 0 = 1 cut lemma eqSymmetry modus ponens metavar var x end metavar ^ 0 = 1 conclude 1 = metavar var x end metavar ^ 0 cut lemma 0<1 conclude not0 0 <= 1 imply not0 not0 0 = 1 cut lemma subLessRight modus ponens 1 = metavar var x end metavar ^ 0 modus ponens not0 0 <= 1 imply not0 not0 0 = 1 conclude not0 0 <= metavar var x end metavar ^ 0 imply not0 not0 0 = metavar var x end metavar ^ 0 end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma positiveBase indu as system Q infer all metavar var m end metavar indeed all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer not0 0 <= metavar var x end metavar ^ metavar var m end metavar imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar imply not0 0 <= metavar var x end metavar ^ metavar var m end metavar + 1 imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar + 1 end define end math ] "

" [ math define proof of lemma positiveBase indu as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer not0 0 <= metavar var x end metavar ^ metavar var m end metavar imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar infer lemma exp(+1) conclude metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar ^ metavar var m end metavar + 1 = metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar conclude metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var m end metavar + 1 cut lemma positiveFactors modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar modus ponens not0 0 <= metavar var x end metavar ^ metavar var m end metavar imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar conclude not0 0 <= metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar imply not0 not0 0 = metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar cut lemma subLessRight modus ponens metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar = metavar var x end metavar ^ metavar var m end metavar + 1 modus ponens not0 0 <= metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar imply not0 not0 0 = metavar var x end metavar * metavar var x end metavar ^ metavar var m end metavar conclude not0 0 <= metavar var x end metavar ^ metavar var m end metavar + 1 imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar + 1 cut all metavar var m end metavar indeed all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer not0 0 <= metavar var x end metavar ^ metavar var m end metavar imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar infer not0 0 <= metavar var x end metavar ^ metavar var m end metavar + 1 imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar + 1 conclude not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar imply not0 0 <= metavar var x end metavar ^ metavar var m end metavar imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar imply not0 0 <= metavar var x end metavar ^ metavar var m end metavar + 1 imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar + 1 cut not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer 1rule mp modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar imply not0 0 <= metavar var x end metavar ^ metavar var m end metavar imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar imply not0 0 <= metavar var x end metavar ^ metavar var m end metavar + 1 imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar + 1 modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude not0 0 <= metavar var x end metavar ^ metavar var m end metavar imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar imply not0 0 <= metavar var x end metavar ^ metavar var m end metavar + 1 imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar + 1 end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma positiveBase as system Q infer all metavar var m end metavar indeed all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer not0 0 <= metavar var x end metavar ^ metavar var m end metavar imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar end define end math ] "

" [ math define proof of lemma positiveBase as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer lemma positiveBase base conclude not0 0 <= metavar var x end metavar ^ 0 imply not0 not0 0 = metavar var x end metavar ^ 0 cut lemma positiveBase indu modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude not0 0 <= metavar var x end metavar ^ metavar var m end metavar imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar imply not0 0 <= metavar var x end metavar ^ metavar var m end metavar + 1 imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar + 1 cut lemma induction modus ponens not0 0 <= metavar var x end metavar ^ 0 imply not0 not0 0 = metavar var x end metavar ^ 0 modus ponens not0 0 <= metavar var x end metavar ^ metavar var m end metavar imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar imply not0 0 <= metavar var x end metavar ^ metavar var m end metavar + 1 imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar + 1 conclude not0 0 <= metavar var x end metavar ^ metavar var m end metavar imply not0 not0 0 = metavar var x end metavar ^ metavar var m end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma positiveToRight(Eq)(1 term) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer 0 = metavar var y end metavar + - metavar var x end metavar end define end math ] "

" [ math define proof of lemma positiveToRight(Eq)(1 term) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer lemma eqSymmetry modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var y end metavar = metavar var x end metavar cut lemma positiveToLeft(Eq)(1 term) modus ponens metavar var y end metavar = metavar var x end metavar conclude metavar var y end metavar + - metavar var x end metavar = 0 cut lemma eqSymmetry modus ponens metavar var y end metavar + - metavar var x end metavar = 0 conclude 0 = metavar var y end metavar + - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

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" [ math define statement of lemma base(1/2)Sum zero exact as system Q infer all metavar var m end metavar indeed base(1/2)Sum( metavar var m end metavar , 0 ) = 1/ 1 + 1 ^ metavar var m end metavar end define end math ] "

" [ math define proof of lemma base(1/2)Sum zero exact as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed lemma eqReflexivity conclude 0 = 0 cut 1rule base(1/2)Sum zero modus ponens 0 = 0 conclude base(1/2)Sum( metavar var m end metavar , 0 ) = 1/ 1 + 1 ^ metavar var m end metavar end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma sameBase(1/2)Sum second base as system Q infer all metavar var m end metavar indeed all metavar var n2 end metavar indeed for all objects metavar var n2 end metavar indeed 0 = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , 0 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) end define end math ] "

" [ math define proof of lemma sameBase(1/2)Sum second base as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n2 end metavar indeed 0 = metavar var n2 end metavar infer lemma eqSymmetry modus ponens 0 = metavar var n2 end metavar conclude metavar var n2 end metavar = 0 cut 1rule base(1/2)Sum zero modus ponens metavar var n2 end metavar = 0 conclude base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) = 1/ 1 + 1 ^ metavar var m end metavar cut lemma eqSymmetry modus ponens base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) = 1/ 1 + 1 ^ metavar var m end metavar conclude 1/ 1 + 1 ^ metavar var m end metavar = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) cut lemma base(1/2)Sum zero exact conclude base(1/2)Sum( metavar var m end metavar , 0 ) = 1/ 1 + 1 ^ metavar var m end metavar cut lemma eqTransitivity modus ponens base(1/2)Sum( metavar var m end metavar , 0 ) = 1/ 1 + 1 ^ metavar var m end metavar modus ponens 1/ 1 + 1 ^ metavar var m end metavar = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) conclude base(1/2)Sum( metavar var m end metavar , 0 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) cut all metavar var m end metavar indeed all metavar var n2 end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n2 end metavar indeed 0 = metavar var n2 end metavar infer base(1/2)Sum( metavar var m end metavar , 0 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) conclude 0 = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , 0 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) cut 1rule gen modus ponens 0 = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , 0 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) conclude for all objects metavar var n2 end metavar indeed 0 = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , 0 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma sameBase(1/2)Sum second indu as system Q infer all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) imply for all objects metavar var n2 end metavar indeed metavar var n1 end metavar + 1 = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) end define end math ] "

" [ math define proof of lemma sameBase(1/2)Sum second indu as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) infer metavar var n1 end metavar + 1 = metavar var n2 end metavar infer lemma +1IsPositive(N) conclude not0 0 <= metavar var n1 end metavar + 1 imply not0 not0 0 = metavar var n1 end metavar + 1 cut 1rule base(1/2)Sum positive modus ponens not0 0 <= metavar var n1 end metavar + 1 imply not0 not0 0 = metavar var n1 end metavar + 1 conclude base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 ) = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n1 end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) cut lemma x=x+y-y conclude metavar var n1 end metavar = metavar var n1 end metavar + 1 + - 1 cut lemma a4 at metavar var n1 end metavar + 1 + - 1 modus ponens for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) conclude metavar var n1 end metavar = metavar var n1 end metavar + 1 + - 1 imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) cut 1rule mp modus ponens metavar var n1 end metavar = metavar var n1 end metavar + 1 + - 1 imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) modus ponens metavar var n1 end metavar = metavar var n1 end metavar + 1 + - 1 conclude base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) cut lemma eqSymmetry modus ponens base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) conclude base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) cut lemma eqAdditionLeft modus ponens base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) conclude 1/ 1 + 1 ^ metavar var m end metavar + metavar var n1 end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n1 end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) cut lemma positiveToRight(Eq) modus ponens metavar var n1 end metavar + 1 = metavar var n2 end metavar conclude metavar var n1 end metavar = metavar var n2 end metavar + - 1 cut lemma a4 at metavar var n2 end metavar + - 1 modus ponens for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) conclude metavar var n1 end metavar = metavar var n2 end metavar + - 1 imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar + - 1 ) cut 1rule mp modus ponens metavar var n1 end metavar = metavar var n2 end metavar + - 1 imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar + - 1 ) modus ponens metavar var n1 end metavar = metavar var n2 end metavar + - 1 conclude base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar + - 1 ) cut lemma eqAdditionLeft modus ponens base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar + - 1 ) conclude 1/ 1 + 1 ^ metavar var m end metavar + metavar var n1 end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n1 end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar + - 1 ) cut lemma eqAdditionLeft modus ponens metavar var n1 end metavar + 1 = metavar var n2 end metavar conclude metavar var m end metavar + metavar var n1 end metavar + 1 = metavar var m end metavar + metavar var n2 end metavar cut lemma sameExp modus ponens metavar var m end metavar + metavar var n1 end metavar + 1 = metavar var m end metavar + metavar var n2 end metavar conclude 1/ 1 + 1 ^ metavar var m end metavar + metavar var n1 end metavar + 1 = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n2 end metavar cut lemma eqAddition modus ponens 1/ 1 + 1 ^ metavar var m end metavar + metavar var n1 end metavar + 1 = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n2 end metavar conclude 1/ 1 + 1 ^ metavar var m end metavar + metavar var n1 end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar + - 1 ) = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n2 end metavar + base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar + - 1 ) cut lemma subLessRight modus ponens metavar var n1 end metavar + 1 = metavar var n2 end metavar modus ponens not0 0 <= metavar var n1 end metavar + 1 imply not0 not0 0 = metavar var n1 end metavar + 1 conclude not0 0 <= metavar var n2 end metavar imply not0 not0 0 = metavar var n2 end metavar cut 1rule base(1/2)Sum positive modus ponens not0 0 <= metavar var n2 end metavar imply not0 not0 0 = metavar var n2 end metavar conclude base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n2 end metavar + base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar + - 1 ) cut lemma eqSymmetry modus ponens base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n2 end metavar + base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar + - 1 ) conclude 1/ 1 + 1 ^ metavar var m end metavar + metavar var n2 end metavar + base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar + - 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) cut lemma eqTransitivity6 modus ponens base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 ) = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n1 end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) modus ponens 1/ 1 + 1 ^ metavar var m end metavar + metavar var n1 end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n1 end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) modus ponens 1/ 1 + 1 ^ metavar var m end metavar + metavar var n1 end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n1 end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar + - 1 ) modus ponens 1/ 1 + 1 ^ metavar var m end metavar + metavar var n1 end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar + - 1 ) = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n2 end metavar + base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar + - 1 ) modus ponens 1/ 1 + 1 ^ metavar var m end metavar + metavar var n2 end metavar + base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar + - 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) conclude base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) cut 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) infer metavar var n1 end metavar + 1 = metavar var n2 end metavar infer base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) conclude for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) imply metavar var n1 end metavar + 1 = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) cut for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) infer 1rule mp modus ponens for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) imply metavar var n1 end metavar + 1 = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) modus ponens for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) conclude metavar var n1 end metavar + 1 = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) cut 1rule gen modus ponens metavar var n1 end metavar + 1 = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) conclude for all objects metavar var n2 end metavar indeed metavar var n1 end metavar + 1 = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) cut all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) infer for all objects metavar var n2 end metavar indeed metavar var n1 end metavar + 1 = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) conclude for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) imply for all objects metavar var n2 end metavar indeed metavar var n1 end metavar + 1 = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma sameBase(1/2)Sum second as system Q infer all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar infer base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) end define end math ] "

" [ math define proof of lemma sameBase(1/2)Sum second as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar infer lemma sameBase(1/2)Sum second base conclude for all objects metavar var n2 end metavar indeed 0 = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , 0 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) cut lemma sameBase(1/2)Sum second indu conclude for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) imply for all objects metavar var n2 end metavar indeed metavar var n1 end metavar + 1 = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) cut lemma induction modus ponens for all objects metavar var n2 end metavar indeed 0 = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , 0 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) modus ponens for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) imply for all objects metavar var n2 end metavar indeed metavar var n1 end metavar + 1 = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar + 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) conclude for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) cut lemma a4 at metavar var n2 end metavar modus ponens for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) conclude metavar var n1 end metavar = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) cut 1rule mp modus ponens metavar var n1 end metavar = metavar var n2 end metavar imply base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) modus ponens metavar var n1 end metavar = metavar var n2 end metavar conclude base(1/2)Sum( metavar var m end metavar , metavar var n1 end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n2 end metavar ) end quote state proof state cache var c end expand end define end math ] "

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" [ math define statement of lemma negativeToLeft(Less)(1 term) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 not0 0 = metavar var x end metavar + - metavar var y end metavar infer not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar end define end math ] "

" [ math define proof of lemma negativeToLeft(Less)(1 term) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 not0 0 = metavar var x end metavar + - metavar var y end metavar infer lemma lessAddition modus ponens not0 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 not0 0 = metavar var x end metavar + - metavar var y end metavar conclude not0 0 + metavar var y end metavar <= metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar imply not0 not0 0 + metavar var y end metavar = metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar cut lemma plus0Left conclude 0 + metavar var y end metavar = metavar var y end metavar cut lemma subLessLeft modus ponens 0 + metavar var y end metavar = metavar var y end metavar modus ponens not0 0 + metavar var y end metavar <= metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar imply not0 not0 0 + metavar var y end metavar = metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar conclude not0 metavar var y end metavar <= metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar cut lemma three2threeTerms conclude metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma x=x+y-y conclude metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar modus ponens metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar conclude metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar = metavar var x end metavar cut lemma subLessRight modus ponens metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar = metavar var x end metavar modus ponens not0 metavar var y end metavar <= metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar conclude not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma base(1/2)Sum(+1) as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed base(1/2)Sum( metavar var m end metavar , metavar var n end metavar + 1 ) = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n end metavar ) end define end math ] "


" [ math define proof of lemma base(1/2)Sum(+1) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed lemma +1IsPositive(N) conclude not0 0 <= metavar var n end metavar + 1 imply not0 not0 0 = metavar var n end metavar + 1 cut 1rule base(1/2)Sum positive modus ponens not0 0 <= metavar var n end metavar + 1 imply not0 not0 0 = metavar var n end metavar + 1 conclude base(1/2)Sum( metavar var m end metavar , metavar var n end metavar + 1 ) = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n end metavar + 1 + - 1 ) cut axiom plusAssociativity conclude metavar var m end metavar + metavar var n end metavar + 1 = metavar var m end metavar + metavar var n end metavar + 1 cut lemma sameExp modus ponens metavar var m end metavar + metavar var n end metavar + 1 = metavar var m end metavar + metavar var n end metavar + 1 conclude 1/ 1 + 1 ^ metavar var m end metavar + metavar var n end metavar + 1 = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n end metavar + 1 cut lemma eqSymmetry modus ponens 1/ 1 + 1 ^ metavar var m end metavar + metavar var n end metavar + 1 = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n end metavar + 1 conclude 1/ 1 + 1 ^ metavar var m end metavar + metavar var n end metavar + 1 = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n end metavar + 1 cut lemma x=x+y-y conclude metavar var n end metavar = metavar var n end metavar + 1 + - 1 cut lemma sameBase(1/2)Sum second modus ponens metavar var n end metavar = metavar var n end metavar + 1 + - 1 conclude base(1/2)Sum( metavar var m end metavar , metavar var n end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n end metavar + 1 + - 1 ) cut lemma eqSymmetry modus ponens base(1/2)Sum( metavar var m end metavar , metavar var n end metavar ) = base(1/2)Sum( metavar var m end metavar , metavar var n end metavar + 1 + - 1 ) conclude base(1/2)Sum( metavar var m end metavar , metavar var n end metavar + 1 + - 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n end metavar ) cut lemma addEquations modus ponens 1/ 1 + 1 ^ metavar var m end metavar + metavar var n end metavar + 1 = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n end metavar + 1 modus ponens base(1/2)Sum( metavar var m end metavar , metavar var n end metavar + 1 + - 1 ) = base(1/2)Sum( metavar var m end metavar , metavar var n end metavar ) conclude 1/ 1 + 1 ^ metavar var m end metavar + metavar var n end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n end metavar + 1 + - 1 ) = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n end metavar ) cut lemma eqTransitivity modus ponens base(1/2)Sum( metavar var m end metavar , metavar var n end metavar + 1 ) = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n end metavar + 1 + - 1 ) modus ponens 1/ 1 + 1 ^ metavar var m end metavar + metavar var n end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n end metavar + 1 + - 1 ) = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n end metavar ) conclude base(1/2)Sum( metavar var m end metavar , metavar var n end metavar + 1 ) = 1/ 1 + 1 ^ metavar var m end metavar + metavar var n end metavar + 1 + base(1/2)Sum( metavar var m end metavar , metavar var n end metavar ) end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma base(1/2)Sum exact bound base as system Q infer all metavar var m end metavar indeed base(1/2)Sum( metavar var m end metavar + 1 , 0 ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + 0 = 1/ 1 + 1 ^ metavar var m end metavar end define end math ] "

" [ math define proof of lemma base(1/2)Sum exact bound base as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed lemma base(1/2)Sum zero exact conclude base(1/2)Sum( metavar var m end metavar + 1 , 0 ) = 1/ 1 + 1 ^ metavar var m end metavar + 1 cut lemma exp(+1) conclude 1/ 1 + 1 ^ metavar var m end metavar + 1 = 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar cut lemma eqTransitivity modus ponens base(1/2)Sum( metavar var m end metavar + 1 , 0 ) = 1/ 1 + 1 ^ metavar var m end metavar + 1 modus ponens 1/ 1 + 1 ^ metavar var m end metavar + 1 = 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar conclude base(1/2)Sum( metavar var m end metavar + 1 , 0 ) = 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar cut axiom plus0 conclude metavar var m end metavar + 1 + 0 = metavar var m end metavar + 1 cut lemma sameExp modus ponens metavar var m end metavar + 1 + 0 = metavar var m end metavar + 1 conclude 1/ 1 + 1 ^ metavar var m end metavar + 1 + 0 = 1/ 1 + 1 ^ metavar var m end metavar + 1 cut lemma eqTransitivity modus ponens 1/ 1 + 1 ^ metavar var m end metavar + 1 + 0 = 1/ 1 + 1 ^ metavar var m end metavar + 1 modus ponens 1/ 1 + 1 ^ metavar var m end metavar + 1 = 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar conclude 1/ 1 + 1 ^ metavar var m end metavar + 1 + 0 = 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar cut lemma addEquations modus ponens base(1/2)Sum( metavar var m end metavar + 1 , 0 ) = 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar modus ponens 1/ 1 + 1 ^ metavar var m end metavar + 1 + 0 = 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar conclude base(1/2)Sum( metavar var m end metavar + 1 , 0 ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + 0 = 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar + 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar cut lemma (1/2)x+(1/2)x=x conclude 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar + 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar = 1/ 1 + 1 ^ metavar var m end metavar cut lemma eqTransitivity modus ponens base(1/2)Sum( metavar var m end metavar + 1 , 0 ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + 0 = 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar + 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar modus ponens 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar + 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar = 1/ 1 + 1 ^ metavar var m end metavar conclude base(1/2)Sum( metavar var m end metavar + 1 , 0 ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + 0 = 1/ 1 + 1 ^ metavar var m end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma base(1/2)Sum exact bound indu as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar = 1/ 1 + 1 ^ metavar var m end metavar imply base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar + 1 ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = 1/ 1 + 1 ^ metavar var m end metavar end define end math ] "

" [ math define proof of lemma base(1/2)Sum exact bound indu as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar = 1/ 1 + 1 ^ metavar var m end metavar infer lemma base(1/2)Sum(+1) conclude base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar + 1 ) = 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 + base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) cut lemma eqAddition modus ponens base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar + 1 ) = 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 + base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) conclude base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar + 1 ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 + base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 cut axiom plusCommutativity conclude 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 + base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) = base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 cut lemma eqAddition modus ponens 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 + base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) = base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 conclude 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 + base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 cut lemma exp(+1) conclude 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar cut lemma addEquations modus ponens 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar modus ponens 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar conclude 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar cut lemma (1/2)x+(1/2)x=x conclude 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar = 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar cut lemma eqTransitivity modus ponens 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar modus ponens 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1/ 1 + 1 * 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar = 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar conclude 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar cut lemma three2twoTerms modus ponens 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar conclude base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar cut lemma eqTransitivity5 modus ponens base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar + 1 ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 + base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 modus ponens 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 + base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 modus ponens base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar modus ponens base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar = 1/ 1 + 1 ^ metavar var m end metavar conclude base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar + 1 ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = 1/ 1 + 1 ^ metavar var m end metavar cut all metavar var m end metavar indeed all metavar var n end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n end metavar indeed base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar = 1/ 1 + 1 ^ metavar var m end metavar infer base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar + 1 ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = 1/ 1 + 1 ^ metavar var m end metavar conclude base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar = 1/ 1 + 1 ^ metavar var m end metavar imply base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar + 1 ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = 1/ 1 + 1 ^ metavar var m end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma base(1/2)Sum exact bound as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar = 1/ 1 + 1 ^ metavar var m end metavar end define end math ] "


" [ math define proof of lemma base(1/2)Sum exact bound as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed lemma base(1/2)Sum exact bound base conclude base(1/2)Sum( metavar var m end metavar + 1 , 0 ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + 0 = 1/ 1 + 1 ^ metavar var m end metavar cut lemma base(1/2)Sum exact bound indu conclude base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar = 1/ 1 + 1 ^ metavar var m end metavar imply base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar + 1 ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = 1/ 1 + 1 ^ metavar var m end metavar cut lemma induction modus ponens base(1/2)Sum( metavar var m end metavar + 1 , 0 ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + 0 = 1/ 1 + 1 ^ metavar var m end metavar modus ponens base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar = 1/ 1 + 1 ^ metavar var m end metavar imply base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar + 1 ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + 1 = 1/ 1 + 1 ^ metavar var m end metavar conclude base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar = 1/ 1 + 1 ^ metavar var m end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma base(1/2)Sum bound as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed not0 base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) <= 1/ 1 + 1 ^ metavar var m end metavar imply not0 not0 base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) = 1/ 1 + 1 ^ metavar var m end metavar end define end math ] "

" [ math define proof of lemma base(1/2)Sum bound as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed lemma base(1/2)Sum exact bound conclude base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar = 1/ 1 + 1 ^ metavar var m end metavar cut axiom plusCommutativity conclude 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) = base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar cut lemma eqTransitivity modus ponens 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) = base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar modus ponens base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) + 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar = 1/ 1 + 1 ^ metavar var m end metavar conclude 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) = 1/ 1 + 1 ^ metavar var m end metavar cut lemma positiveToRight(Eq) modus ponens 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar + base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) = 1/ 1 + 1 ^ metavar var m end metavar conclude 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar = 1/ 1 + 1 ^ metavar var m end metavar + - base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) cut lemma 0<1/2 conclude not0 0 <= 1/ 1 + 1 imply not0 not0 0 = 1/ 1 + 1 cut lemma positiveBase modus ponens not0 0 <= 1/ 1 + 1 imply not0 not0 0 = 1/ 1 + 1 conclude not0 0 <= 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar imply not0 not0 0 = 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar cut lemma subLessRight modus ponens 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar = 1/ 1 + 1 ^ metavar var m end metavar + - base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) modus ponens not0 0 <= 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar imply not0 not0 0 = 1/ 1 + 1 ^ metavar var m end metavar + 1 + metavar var n end metavar conclude not0 0 <= 1/ 1 + 1 ^ metavar var m end metavar + - base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) imply not0 not0 0 = 1/ 1 + 1 ^ metavar var m end metavar + - base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) cut lemma negativeToLeft(Less)(1 term) modus ponens not0 0 <= 1/ 1 + 1 ^ metavar var m end metavar + - base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) imply not0 not0 0 = 1/ 1 + 1 ^ metavar var m end metavar + - base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) conclude not0 base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) <= 1/ 1 + 1 ^ metavar var m end metavar imply not0 not0 base(1/2)Sum( metavar var m end metavar + 1 , metavar var n end metavar ) = 1/ 1 + 1 ^ metavar var m end metavar end quote state proof state cache var c end expand end define end math ] "

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" [ math define statement of lemma sameSeries(NumDiff) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var o end metavar indeed all metavar var p end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed metavar var o end metavar = metavar var p end metavar infer metavar var n1 end metavar = metavar var n2 end metavar infer | [ metavar var fx end metavar ; metavar var o end metavar ] + - [ metavar var fy end metavar ; metavar var n1 end metavar ] | = | [ metavar var fx end metavar ; metavar var p end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | end define end math ] "


" [ math define proof of lemma sameSeries(NumDiff) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var o end metavar indeed all metavar var p end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed metavar var o end metavar = metavar var p end metavar infer metavar var n1 end metavar = metavar var n2 end metavar infer lemma sameSeries modus ponens metavar var o end metavar = metavar var p end metavar conclude [ metavar var fx end metavar ; metavar var o end metavar ] = [ metavar var fx end metavar ; metavar var p end metavar ] cut lemma sameSeries modus ponens metavar var n1 end metavar = metavar var n2 end metavar conclude [ metavar var fy end metavar ; metavar var n1 end metavar ] = [ metavar var fy end metavar ; metavar var n2 end metavar ] cut lemma eqNegated modus ponens [ metavar var fy end metavar ; metavar var n1 end metavar ] = [ metavar var fy end metavar ; metavar var n2 end metavar ] conclude - [ metavar var fy end metavar ; metavar var n1 end metavar ] = - [ metavar var fy end metavar ; metavar var n2 end metavar ] cut lemma addEquations modus ponens [ metavar var fx end metavar ; metavar var o end metavar ] = [ metavar var fx end metavar ; metavar var p end metavar ] modus ponens - [ metavar var fy end metavar ; metavar var n1 end metavar ] = - [ metavar var fy end metavar ; metavar var n2 end metavar ] conclude [ metavar var fx end metavar ; metavar var o end metavar ] + - [ metavar var fy end metavar ; metavar var n1 end metavar ] = [ metavar var fx end metavar ; metavar var p end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] cut lemma sameNumerical modus ponens [ metavar var fx end metavar ; metavar var o end metavar ] + - [ metavar var fy end metavar ; metavar var n1 end metavar ] = [ metavar var fx end metavar ; metavar var p end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] conclude | [ metavar var fx end metavar ; metavar var o end metavar ] + - [ metavar var fy end metavar ; metavar var n1 end metavar ] | = | [ metavar var fx end metavar ; metavar var p end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma UStelescope zero exact as system Q infer all metavar var m end metavar indeed UStelescope( metavar var m end metavar , 0 ) = | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 1 ] | end define end math ] "

" [ math define proof of lemma UStelescope zero exact as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed lemma eqReflexivity conclude 0 = 0 cut 1rule UStelescope zero modus ponens 0 = 0 conclude UStelescope( metavar var m end metavar , 0 ) = | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 1 ] | end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma sameTelescope second base as system Q infer all metavar var m end metavar indeed all metavar var n2 end metavar indeed for all objects metavar var n2 end metavar indeed 0 = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , 0 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) end define end math ] "

" [ math define proof of lemma sameTelescope second base as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n2 end metavar indeed 0 = metavar var n2 end metavar infer lemma eqSymmetry modus ponens 0 = metavar var n2 end metavar conclude metavar var n2 end metavar = 0 cut 1rule UStelescope zero modus ponens metavar var n2 end metavar = 0 conclude UStelescope( metavar var m end metavar , metavar var n2 end metavar ) = | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 1 ] | cut lemma eqSymmetry modus ponens UStelescope( metavar var m end metavar , metavar var n2 end metavar ) = | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 1 ] | conclude | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 1 ] | = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) cut lemma UStelescope zero exact conclude UStelescope( metavar var m end metavar , 0 ) = | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 1 ] | cut lemma eqTransitivity modus ponens UStelescope( metavar var m end metavar , 0 ) = | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 1 ] | modus ponens | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 1 ] | = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) conclude UStelescope( metavar var m end metavar , 0 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) cut all metavar var m end metavar indeed all metavar var n2 end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n2 end metavar indeed 0 = metavar var n2 end metavar infer UStelescope( metavar var m end metavar , 0 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) conclude 0 = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , 0 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) cut 1rule gen modus ponens 0 = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , 0 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) conclude for all objects metavar var n2 end metavar indeed 0 = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , 0 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma sameTelescope second indu as system Q infer all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) imply for all objects metavar var n2 end metavar indeed metavar var n1 end metavar + 1 = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) end define end math ] "

" [ math define proof of lemma sameTelescope second indu as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) infer metavar var n1 end metavar + 1 = metavar var n2 end metavar infer lemma +1IsPositive(N) conclude not0 0 <= metavar var n1 end metavar + 1 imply not0 not0 0 = metavar var n1 end metavar + 1 cut 1rule UStelescope positive modus ponens not0 0 <= metavar var n1 end metavar + 1 imply not0 not0 0 = metavar var n1 end metavar + 1 conclude UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 ) = | [ us ; metavar var m end metavar + metavar var n1 end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n1 end metavar + 1 + 1 ] | + UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) cut lemma x=x+y-y conclude metavar var n1 end metavar = metavar var n1 end metavar + 1 + - 1 cut lemma a4 at metavar var n1 end metavar + 1 + - 1 modus ponens for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) conclude metavar var n1 end metavar = metavar var n1 end metavar + 1 + - 1 imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) cut 1rule mp modus ponens metavar var n1 end metavar = metavar var n1 end metavar + 1 + - 1 imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) modus ponens metavar var n1 end metavar = metavar var n1 end metavar + 1 + - 1 conclude UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) cut lemma eqSymmetry modus ponens UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) conclude UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) = UStelescope( metavar var m end metavar , metavar var n1 end metavar ) cut lemma positiveToRight(Eq) modus ponens metavar var n1 end metavar + 1 = metavar var n2 end metavar conclude metavar var n1 end metavar = metavar var n2 end metavar + - 1 cut lemma a4 at metavar var n2 end metavar + - 1 modus ponens for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) conclude metavar var n1 end metavar = metavar var n2 end metavar + - 1 imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar + - 1 ) cut 1rule mp modus ponens metavar var n1 end metavar = metavar var n2 end metavar + - 1 imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar + - 1 ) modus ponens metavar var n1 end metavar = metavar var n2 end metavar + - 1 conclude UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar + - 1 ) cut lemma eqTransitivity modus ponens UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) = UStelescope( metavar var m end metavar , metavar var n1 end metavar ) modus ponens UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar + - 1 ) conclude UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar + - 1 ) cut lemma eqAdditionLeft modus ponens metavar var n1 end metavar + 1 = metavar var n2 end metavar conclude metavar var m end metavar + metavar var n1 end metavar + 1 = metavar var m end metavar + metavar var n2 end metavar cut lemma eqAddition modus ponens metavar var n1 end metavar + 1 = metavar var n2 end metavar conclude metavar var n1 end metavar + 1 + 1 = metavar var n2 end metavar + 1 cut lemma eqAdditionLeft modus ponens metavar var n1 end metavar + 1 + 1 = metavar var n2 end metavar + 1 conclude metavar var m end metavar + metavar var n1 end metavar + 1 + 1 = metavar var m end metavar + metavar var n2 end metavar + 1 cut lemma sameSeries(NumDiff) modus ponens metavar var m end metavar + metavar var n1 end metavar + 1 = metavar var m end metavar + metavar var n2 end metavar modus ponens metavar var m end metavar + metavar var n1 end metavar + 1 + 1 = metavar var m end metavar + metavar var n2 end metavar + 1 conclude | [ us ; metavar var m end metavar + metavar var n1 end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n1 end metavar + 1 + 1 ] | = | [ us ; metavar var m end metavar + metavar var n2 end metavar ] + - [ us ; metavar var m end metavar + metavar var n2 end metavar + 1 ] | cut lemma addEquations modus ponens | [ us ; metavar var m end metavar + metavar var n1 end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n1 end metavar + 1 + 1 ] | = | [ us ; metavar var m end metavar + metavar var n2 end metavar ] + - [ us ; metavar var m end metavar + metavar var n2 end metavar + 1 ] | modus ponens UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar + - 1 ) conclude | [ us ; metavar var m end metavar + metavar var n1 end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n1 end metavar + 1 + 1 ] | + UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) = | [ us ; metavar var m end metavar + metavar var n2 end metavar ] + - [ us ; metavar var m end metavar + metavar var n2 end metavar + 1 ] | + UStelescope( metavar var m end metavar , metavar var n2 end metavar + - 1 ) cut lemma subLessRight modus ponens metavar var n1 end metavar + 1 = metavar var n2 end metavar modus ponens not0 0 <= metavar var n1 end metavar + 1 imply not0 not0 0 = metavar var n1 end metavar + 1 conclude not0 0 <= metavar var n2 end metavar imply not0 not0 0 = metavar var n2 end metavar cut 1rule UStelescope positive modus ponens not0 0 <= metavar var n2 end metavar imply not0 not0 0 = metavar var n2 end metavar conclude UStelescope( metavar var m end metavar , metavar var n2 end metavar ) = | [ us ; metavar var m end metavar + metavar var n2 end metavar ] + - [ us ; metavar var m end metavar + metavar var n2 end metavar + 1 ] | + UStelescope( metavar var m end metavar , metavar var n2 end metavar + - 1 ) cut lemma eqSymmetry modus ponens UStelescope( metavar var m end metavar , metavar var n2 end metavar ) = | [ us ; metavar var m end metavar + metavar var n2 end metavar ] + - [ us ; metavar var m end metavar + metavar var n2 end metavar + 1 ] | + UStelescope( metavar var m end metavar , metavar var n2 end metavar + - 1 ) conclude | [ us ; metavar var m end metavar + metavar var n2 end metavar ] + - [ us ; metavar var m end metavar + metavar var n2 end metavar + 1 ] | + UStelescope( metavar var m end metavar , metavar var n2 end metavar + - 1 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) cut lemma eqTransitivity4 modus ponens UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 ) = | [ us ; metavar var m end metavar + metavar var n1 end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n1 end metavar + 1 + 1 ] | + UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) modus ponens | [ us ; metavar var m end metavar + metavar var n1 end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n1 end metavar + 1 + 1 ] | + UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 + - 1 ) = | [ us ; metavar var m end metavar + metavar var n2 end metavar ] + - [ us ; metavar var m end metavar + metavar var n2 end metavar + 1 ] | + UStelescope( metavar var m end metavar , metavar var n2 end metavar + - 1 ) modus ponens | [ us ; metavar var m end metavar + metavar var n2 end metavar ] + - [ us ; metavar var m end metavar + metavar var n2 end metavar + 1 ] | + UStelescope( metavar var m end metavar , metavar var n2 end metavar + - 1 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) conclude UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) cut 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) infer metavar var n1 end metavar + 1 = metavar var n2 end metavar infer UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) conclude for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) imply metavar var n1 end metavar + 1 = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) cut for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) infer 1rule mp modus ponens for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) imply metavar var n1 end metavar + 1 = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) modus ponens for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) conclude metavar var n1 end metavar + 1 = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) cut 1rule gen modus ponens metavar var n1 end metavar + 1 = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) conclude for all objects metavar var n2 end metavar indeed metavar var n1 end metavar + 1 = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) cut all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) infer for all objects metavar var n2 end metavar indeed metavar var n1 end metavar + 1 = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) conclude for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) imply for all objects metavar var n2 end metavar indeed metavar var n1 end metavar + 1 = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma sameTelescope second as system Q infer all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar infer UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) end define end math ] "


" [ math define proof of lemma sameTelescope second as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar infer lemma sameTelescope second base conclude for all objects metavar var n2 end metavar indeed 0 = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , 0 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) cut lemma sameTelescope second indu conclude for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) imply for all objects metavar var n2 end metavar indeed metavar var n1 end metavar + 1 = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) cut lemma induction modus ponens for all objects metavar var n2 end metavar indeed 0 = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , 0 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) modus ponens for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) imply for all objects metavar var n2 end metavar indeed metavar var n1 end metavar + 1 = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar + 1 ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) conclude for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) cut lemma a4 at metavar var n2 end metavar modus ponens for all objects metavar var n2 end metavar indeed metavar var n1 end metavar = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) conclude metavar var n1 end metavar = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) cut 1rule mp modus ponens metavar var n1 end metavar = metavar var n2 end metavar imply UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) modus ponens metavar var n1 end metavar = metavar var n2 end metavar conclude UStelescope( metavar var m end metavar , metavar var n1 end metavar ) = UStelescope( metavar var m end metavar , metavar var n2 end metavar ) end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma telescopeNumerical base as system Q infer all metavar var m end metavar indeed | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 0 + 1 ] | <= UStelescope( metavar var m end metavar , 0 ) end define end math ] "

" [ math define proof of lemma telescopeNumerical base as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed lemma eqReflexivity conclude 0 = 0 cut 1rule UStelescope zero modus ponens 0 = 0 conclude UStelescope( metavar var m end metavar , 0 ) = | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 1 ] | cut lemma eqReflexivity conclude metavar var m end metavar = metavar var m end metavar cut lemma plus0Left conclude 0 + 1 = 1 cut lemma eqAdditionLeft modus ponens 0 + 1 = 1 conclude metavar var m end metavar + 0 + 1 = metavar var m end metavar + 1 cut lemma eqSymmetry modus ponens metavar var m end metavar + 0 + 1 = metavar var m end metavar + 1 conclude metavar var m end metavar + 1 = metavar var m end metavar + 0 + 1 cut lemma sameSeries(NumDiff) modus ponens metavar var m end metavar = metavar var m end metavar modus ponens metavar var m end metavar + 1 = metavar var m end metavar + 0 + 1 conclude | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 1 ] | = | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 0 + 1 ] | cut lemma eqTransitivity modus ponens UStelescope( metavar var m end metavar , 0 ) = | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 1 ] | modus ponens | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 1 ] | = | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 0 + 1 ] | conclude UStelescope( metavar var m end metavar , 0 ) = | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 0 + 1 ] | cut lemma eqSymmetry modus ponens UStelescope( metavar var m end metavar , 0 ) = | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 0 + 1 ] | conclude | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 0 + 1 ] | = UStelescope( metavar var m end metavar , 0 ) cut lemma eqLeq modus ponens | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 0 + 1 ] | = UStelescope( metavar var m end metavar , 0 ) conclude | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 0 + 1 ] | <= UStelescope( metavar var m end metavar , 0 ) end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma telescopeNumerical indu as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar ) imply | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar + 1 ) end define end math ] "

" [ math define proof of lemma telescopeNumerical indu as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar ) infer lemma +1IsPositive(N) conclude not0 0 <= metavar var n end metavar + 1 imply not0 not0 0 = metavar var n end metavar + 1 cut 1rule UStelescope positive modus ponens not0 0 <= metavar var n end metavar + 1 imply not0 not0 0 = metavar var n end metavar + 1 conclude UStelescope( metavar var m end metavar , metavar var n end metavar + 1 ) = | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + UStelescope( metavar var m end metavar , metavar var n end metavar + 1 + - 1 ) cut lemma x=x+y-y conclude metavar var n end metavar = metavar var n end metavar + 1 + - 1 cut lemma eqSymmetry modus ponens metavar var n end metavar = metavar var n end metavar + 1 + - 1 conclude metavar var n end metavar + 1 + - 1 = metavar var n end metavar cut lemma sameTelescope second modus ponens metavar var n end metavar + 1 + - 1 = metavar var n end metavar conclude UStelescope( metavar var m end metavar , metavar var n end metavar + 1 + - 1 ) = UStelescope( metavar var m end metavar , metavar var n end metavar ) cut lemma eqAdditionLeft modus ponens UStelescope( metavar var m end metavar , metavar var n end metavar + 1 + - 1 ) = UStelescope( metavar var m end metavar , metavar var n end metavar ) conclude | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + UStelescope( metavar var m end metavar , metavar var n end metavar + 1 + - 1 ) = | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + UStelescope( metavar var m end metavar , metavar var n end metavar ) cut lemma eqTransitivity modus ponens UStelescope( metavar var m end metavar , metavar var n end metavar + 1 ) = | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + UStelescope( metavar var m end metavar , metavar var n end metavar + 1 + - 1 ) modus ponens | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + UStelescope( metavar var m end metavar , metavar var n end metavar + 1 + - 1 ) = | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + UStelescope( metavar var m end metavar , metavar var n end metavar ) conclude UStelescope( metavar var m end metavar , metavar var n end metavar + 1 ) = | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + UStelescope( metavar var m end metavar , metavar var n end metavar ) cut lemma eqSymmetry modus ponens UStelescope( metavar var m end metavar , metavar var n end metavar + 1 ) = | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + UStelescope( metavar var m end metavar , metavar var n end metavar ) conclude | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + UStelescope( metavar var m end metavar , metavar var n end metavar ) = UStelescope( metavar var m end metavar , metavar var n end metavar + 1 ) cut lemma leqAdditionLeft modus ponens | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar ) conclude | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | <= | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + UStelescope( metavar var m end metavar , metavar var n end metavar ) cut lemma subLeqRight modus ponens | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + UStelescope( metavar var m end metavar , metavar var n end metavar ) = UStelescope( metavar var m end metavar , metavar var n end metavar + 1 ) modus ponens | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | <= | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + UStelescope( metavar var m end metavar , metavar var n end metavar ) conclude | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar + 1 ) cut axiom plusCommutativity conclude | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | = | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | + | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | cut lemma subLeqLeft modus ponens | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | = | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | + | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | modus ponens | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | + | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar + 1 ) conclude | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | + | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar + 1 ) cut lemma insertMiddleTerm(Numerical) conclude | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | <= | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | + | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | cut lemma leqTransitivity modus ponens | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | <= | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | + | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | modus ponens | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | + | [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar + 1 ) conclude | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar + 1 ) cut all metavar var m end metavar indeed all metavar var n end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n end metavar indeed | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar ) infer | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar + 1 ) conclude | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar ) imply | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar + 1 ) end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma telescopeNumerical as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar ) end define end math ] "

" [ math define proof of lemma telescopeNumerical as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed lemma telescopeNumerical base conclude | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 0 + 1 ] | <= UStelescope( metavar var m end metavar , 0 ) cut lemma telescopeNumerical indu conclude | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar ) imply | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar + 1 ) cut lemma induction modus ponens | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + 0 + 1 ] | <= UStelescope( metavar var m end metavar , 0 ) modus ponens | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar ) imply | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar + 1 ) conclude | [ us ; metavar var m end metavar ] + - [ us ; metavar var m end metavar + metavar var n end metavar + 1 ] | <= UStelescope( metavar var m end metavar , metavar var n end metavar ) end quote state proof state cache var c end expand end define end math ] "



---------------(21.10.06)

" [ math define statement of lemma eqAdditionLeft(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma eqAdditionLeft(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer 1rule adhoc eqAddition(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plusCommutativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plusCommutativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity4 modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma x=x+(y-y)(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma x=x+(y-y)(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed lemma plus0(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma negative(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqAdditionLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma x=x+y-y(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma x=x+y-y(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed lemma x=x+(y-y)(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plusAssociativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "



----------(22.10.06)


" [ math define statement of lemma three2twoTerms(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var fu end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma three2twoTerms(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var fu end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma eqAdditionLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plusAssociativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma positiveToRight(Less)(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var end define end math ] "

" [ math define proof of lemma positiveToRight(Less)(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer lemma lessAddition(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var cut lemma x=x+y-y(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subLessLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma three2threeTerms(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma three2threeTerms(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed lemma plusCommutativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma three2twoTerms(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plusAssociativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "


--------(22.10.06)

" [ math define statement of lemma plus0Left(R) as system Q infer all metavar var fx end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma plus0Left(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed lemma plus0(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plusCommutativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "






" [ math define statement of lemma positiveToRight(Eq)(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma positiveToRight(Eq)(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer 1rule adhoc eqAddition(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma x=x+y-y(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma subtractEquations(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var fu end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma subtractEquations(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var fu end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer 1rule adhoc eqAddition(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plus0Left(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma positiveToRight(Eq)(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqAdditionLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plusAssociativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plus0(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity4 modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma x=x+y-y(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity4 modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma neqAddition(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma neqAddition(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma ==Reflexivity conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subtractEquations(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma from contradiction modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer 1rule mp modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma imply negation modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "

-----------(22.10.06)

" [ math define statement of lemma positiveToRight(Less)(1 term)(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var end define end math ] "

" [ math define proof of lemma positiveToRight(Less)(1 term)(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer lemma plus0Left(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subLessLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var cut lemma positiveToRight(Less)(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var end quote state proof state cache var c end expand end define end math ] "

-----------(22.10.06)



" [ math define statement of lemma to!!== as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma to!!== as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer 1rule from== modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer prop lemma mt modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var modus ponens not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma switchTerms(x<=y-z) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar infer metavar var z end metavar <= metavar var y end metavar + - metavar var x end metavar end define end math ] "

" [ math define proof of lemma switchTerms(x<=y-z) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar infer lemma negativeToLeft(Leq) modus ponens metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar cut axiom plusCommutativity conclude metavar var x end metavar + metavar var z end metavar = metavar var z end metavar + metavar var x end metavar cut lemma subLeqLeft modus ponens metavar var x end metavar + metavar var z end metavar = metavar var z end metavar + metavar var x end metavar modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar conclude metavar var z end metavar + metavar var x end metavar <= metavar var y end metavar cut lemma positiveToRight(Leq) modus ponens metavar var z end metavar + metavar var x end metavar <= metavar var y end metavar conclude metavar var z end metavar <= metavar var y end metavar + - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "






" [ math define statement of lemma lessNeq(F) helper as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar imply not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | = metavar var ep end metavar end define end math ] "

" [ math define proof of lemma lessNeq(F) helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar infer lemma a4 at metavar var n end metavar modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var n end metavar imply [ metavar var fx end metavar ; metavar var n end metavar ] <= [ metavar var fy end metavar ; metavar var n end metavar ] + - metavar var ep end metavar cut prop lemma first conjunct modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var n end metavar imply [ metavar var fx end metavar ; metavar var n end metavar ] <= [ metavar var fy end metavar ; metavar var n end metavar ] + - metavar var ep end metavar conclude not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar cut prop lemma second conjunct modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var n end metavar imply [ metavar var fx end metavar ; metavar var n end metavar ] <= [ metavar var fy end metavar ; metavar var n end metavar ] + - metavar var ep end metavar conclude metavar var n end metavar <= metavar var n end metavar imply [ metavar var fx end metavar ; metavar var n end metavar ] <= [ metavar var fy end metavar ; metavar var n end metavar ] + - metavar var ep end metavar cut axiom leqReflexivity conclude metavar var n end metavar <= metavar var n end metavar cut 1rule mp modus ponens metavar var n end metavar <= metavar var n end metavar imply [ metavar var fx end metavar ; metavar var n end metavar ] <= [ metavar var fy end metavar ; metavar var n end metavar ] + - metavar var ep end metavar modus ponens metavar var n end metavar <= metavar var n end metavar conclude [ metavar var fx end metavar ; metavar var n end metavar ] <= [ metavar var fy end metavar ; metavar var n end metavar ] + - metavar var ep end metavar cut lemma switchTerms(x<=y-z) modus ponens [ metavar var fx end metavar ; metavar var n end metavar ] <= [ metavar var fy end metavar ; metavar var n end metavar ] + - metavar var ep end metavar conclude metavar var ep end metavar <= [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] cut lemma x<=|x| conclude [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] <= | [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] | cut lemma leqTransitivity modus ponens metavar var ep end metavar <= [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] modus ponens [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] <= | [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] | conclude metavar var ep end metavar <= | [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] | cut lemma numericalDifference conclude | [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] | = | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | cut lemma subLeqRight modus ponens | [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] | = | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | modus ponens metavar var ep end metavar <= | [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] | conclude metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | cut lemma toNotLess modus ponens metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | conclude not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | = metavar var ep end metavar cut prop lemma to negated double imply modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar modus ponens metavar var n end metavar <= metavar var n end metavar modus ponens not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | = metavar var ep end metavar conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | = metavar var ep end metavar cut all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar infer not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | = metavar var ep end metavar conclude for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar imply not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | = metavar var ep end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma lessNeq(F) as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var end define end math ] "

" [ math define proof of lemma lessNeq(F) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var cut 1rule deduction modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects metavar var ep end metavar indeed not0 not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar cut lemma lessNeq(F) helper conclude for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar imply not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | = metavar var ep end metavar cut pred lemma 2exist mp modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar imply not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | = metavar var ep end metavar modus ponens not0 for all objects metavar var ep end metavar indeed not0 not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | = metavar var ep end metavar cut pred lemma intro exist at metavar var n end metavar modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | = metavar var ep end metavar conclude not0 for all objects metavar var m end metavar indeed not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar cut 1rule gen modus ponens not0 for all objects metavar var m end metavar indeed not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar conclude for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar cut pred lemma intro exist at metavar var ep end metavar modus ponens for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar conclude not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar cut pred lemma toNegatedAEA modus ponens not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar conclude not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar cut 1rule deduction modus ponens not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar conclude not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma lessNeq(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma lessNeq(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var cut lemma lessNeq(F) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut lemma to!!== modus ponens not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "





" [ math define statement of lemma positiveToRight(Less)(1 term) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 0 <= metavar var y end metavar + - metavar var x end metavar imply not0 not0 0 = metavar var y end metavar + - metavar var x end metavar end define end math ] "

" [ math define proof of lemma positiveToRight(Less)(1 term) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer lemma plus0Left conclude 0 + metavar var x end metavar = metavar var x end metavar cut lemma eqSymmetry modus ponens 0 + metavar var x end metavar = metavar var x end metavar conclude metavar var x end metavar = 0 + metavar var x end metavar cut lemma subLessLeft modus ponens metavar var x end metavar = 0 + metavar var x end metavar modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 0 + metavar var x end metavar <= metavar var y end metavar imply not0 not0 0 + metavar var x end metavar = metavar var y end metavar cut lemma positiveToRight(Less) modus ponens not0 0 + metavar var x end metavar <= metavar var y end metavar imply not0 not0 0 + metavar var x end metavar = metavar var y end metavar conclude not0 0 <= metavar var y end metavar + - metavar var x end metavar imply not0 not0 0 = metavar var y end metavar + - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma toLeq(Advanced)(R) as system Q infer all metavar var fep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fep end metavar ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma toLeq(Advanced)(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer lemma positiveToRight(Less)(1 term)(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var cut lemma plusCommutativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma three2threeTerms(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma x=x+y-y(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity4 modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma join conjuncts modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut pred lemma intro exist at the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 for all objects metavar var fep end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fep end metavar ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var fep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects metavar var fep end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fep end metavar ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 for all objects metavar var fep end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fep end metavar ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fep end metavar ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer prop lemma to negated and modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fep end metavar ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fep end metavar ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule gen modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fep end metavar ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude for all objects metavar var fep end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fep end metavar ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut pred lemma (A~)to(~E) modus ponens for all objects metavar var fep end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fep end metavar ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects metavar var fep end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fep end metavar ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma mt modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 for all objects metavar var fep end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fep end metavar ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects metavar var fep end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fep end metavar ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fep end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var cut lemma fromNotLess(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "

--------(23.10.06)


" [ math define statement of lemma leqNeqLess(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var end define end math ] "

" [ math define proof of lemma leqNeqLess(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma leqAntisymmetry(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma from contradiction modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer prop lemma mp2 modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma imply negation modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma toLess(R) modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma subLeqLeft(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma subLeqLeft(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqLeq(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma leqTransitivity(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma leqLessTransitivity(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var end define end math ] "

" [ math define proof of lemma leqLessTransitivity(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma lessLeq(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma lessNeq(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subLeqLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma leqAntisymmetry(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma from contradiction modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer prop lemma mp2 modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma imply negation modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma lessLeq(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma leqTransitivity(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma leqNeqLess(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var end quote state proof state cache var c end expand end define end math ] "

-----------(23.10.06)


" [ math define statement of lemma negativeToLeft(Eq)(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma negativeToLeft(Eq)(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer 1rule adhoc eqAddition(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma three2threeTerms(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma x=x+y-y(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity4 modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma negativeToRight(Less)(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var end define end math ] "

" [ math define proof of lemma negativeToRight(Less)(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer lemma lessAddition(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var cut lemma three2threeTerms(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma x=x+y-y(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subLessLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma !!==Symmetry as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma !!==Symmetry as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer prop lemma mt modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "


-------------(23.10.06)


" [ math define statement of lemma negativeToRight(Eq)(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma negativeToRight(Eq)(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer 1rule adhoc eqAddition(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma x=x+y-y(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma three2threeTerms(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity4 modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma negativeToRight(Eq)(1 term)(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma negativeToRight(Eq)(1 term)(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma negativeToRight(Eq)(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plus0Left(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma doubleMinus(R) as system Q infer all metavar var fx end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma doubleMinus(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed lemma negative(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plusCommutativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma negativeToRight(Eq)(1 term)(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "






" [ math define statement of lemma uniqueNegative(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma uniqueNegative(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma plusCommutativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma positiveToRight(Eq)(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plusCommutativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma positiveToRight(Eq)(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "





" [ math define statement of lemma subtractEquationsLeft(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var fu end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma subtractEquationsLeft(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var fu end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma plusCommutativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plusCommutativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity4 modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subtractEquations(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fu end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma eqNegated(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma eqNegated(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma negative(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma negative(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subtractEquationsLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma neqNegated(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma neqNegated(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma eqNegated(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma doubleMinus(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma doubleMinus(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity4 modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma from contradiction modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer 1rule mp modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma imply negation modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "

-----------(23.10.06)




" [ math define statement of lemma subLeqRight(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma subLeqRight(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma eqLeq(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma leqTransitivity(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma leqNegated(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma leqNegated(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma leqAddition(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma negative(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subLeqLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plusCommutativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subLeqRight(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma leqAddition(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plus0Left(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma x=x+y-y(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subLeqLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subLeqRight(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma lessNegated(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var end define end math ] "


" [ math define proof of lemma lessNegated(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer lemma lessLeq(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma leqNegated(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma lessNeq(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma neqNegated(R) modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma !!==Symmetry modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma leqNeqLess(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma -0=0(R) as system Q infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma -0=0(R) as lambda var c dot lambda var x dot proof expand quote system Q infer lemma negative(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plus0(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma uniqueNegative(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma negativeNegated(R) as system Q infer all metavar var fx end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var end define end math ] "

" [ math define proof of lemma negativeNegated(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var infer lemma lessNegated(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var cut lemma -0=0(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subLessLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma leqTotality(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma leqTotality(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma toLess(R) modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var cut lemma lessLeq(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule repetition modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma from leqGeq(R) as system Q infer all metavar var a end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply metavar var a end metavar infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply metavar var a end metavar infer metavar var a end metavar end define end math ] "

" [ math define proof of lemma from leqGeq(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply metavar var a end metavar infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply metavar var a end metavar infer lemma leqTotality(R) conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma from disjuncts modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply metavar var a end metavar modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply metavar var a end metavar conclude metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "

--------(24.10.06)

" [ math define statement of lemma fromLess(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma fromLess(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma lessLeq(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma leqAntisymmetry(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma lessNeq(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma from contradiction modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer 1rule mp modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma imply negation modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma from<<== as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var end define end math ] "

" [ math define proof of lemma from<<== as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer 1rule repetition modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule repetition modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer prop lemma weaken or second modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer 1rule from== conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut prop lemma weaken or first modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut prop lemma from disjuncts modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut 1rule repetition modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma nonnegativeNumerical(F) as system Q infer all metavar var fx end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var infer |f metavar var fx end metavar | = metavar var fx end metavar end define end math ] "

" [ math define proof of lemma nonnegativeNumerical(F) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var infer axiom numericalF conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var imply |f metavar var fx end metavar | = metavar var fx end metavar imply not0 not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var imply |f metavar var fx end metavar | = the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set cut prop lemma first conjunct modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var imply |f metavar var fx end metavar | = metavar var fx end metavar imply not0 not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var imply |f metavar var fx end metavar | = the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var imply |f metavar var fx end metavar | = metavar var fx end metavar cut 1rule mp modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var imply |f metavar var fx end metavar | = metavar var fx end metavar modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var conclude |f metavar var fx end metavar | = metavar var fx end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma nonnegativeNumerical(R) as system Q infer all metavar var fx end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma nonnegativeNumerical(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma from<<== modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var cut lemma nonnegativeNumerical(F) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var conclude |f metavar var fx end metavar | = metavar var fx end metavar cut 1rule adhoc sameR modus ponens |f metavar var fx end metavar | = metavar var fx end metavar conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule repetition modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma to<<== as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma to<<== as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer 1rule repetition modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var cut prop lemma weaken or second modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer 1rule to== modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma weaken or first modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma from disjuncts modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule repetition modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma negativeNumerical(F) as system Q infer all metavar var fx end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var infer |f metavar var fx end metavar | = the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set end define end math ] "

" [ math define proof of lemma negativeNumerical(F) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var infer axiom numericalF conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var imply |f metavar var fx end metavar | = metavar var fx end metavar imply not0 not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var imply |f metavar var fx end metavar | = the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set cut prop lemma second conjunct modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var imply |f metavar var fx end metavar | = metavar var fx end metavar imply not0 not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var imply |f metavar var fx end metavar | = the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var imply |f metavar var fx end metavar | = the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set cut 1rule mp modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var imply |f metavar var fx end metavar | = the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var conclude |f metavar var fx end metavar | = the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma negativeNumerical(R) as system Q infer all metavar var fx end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "


" [ math define proof of lemma negativeNumerical(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var infer lemma fromLess(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma to<<== modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma from contradiction modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var cut all metavar var fx end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var infer not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var imply not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var cut not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var infer 1rule mp modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var imply not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var imply not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var cut prop lemma imply negation modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var imply not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ metavar var fx end metavar ; object var var m end var ] | = object var var ep end var cut lemma negativeNumerical(F) conclude |f metavar var fx end metavar | = the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set cut 1rule adhoc sameR modus ponens |f metavar var fx end metavar | = the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule repetition modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma 0<=|x|(R) as system Q infer all metavar var fx end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma 0<=|x|(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma nonnegativeNumerical(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subLeqRight(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma toLess(R) modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var cut lemma negativeNumerical(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma negativeNegated(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var cut lemma lessLeq(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subLeqRight(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule deduction modus ponens all metavar var fx end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma from negations modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "

-----------(24.10.06)

" [ math define statement of lemma positiveNegated(R) as system Q infer all metavar var fx end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var end define end math ] "

" [ math define proof of lemma positiveNegated(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var infer lemma lessNegated(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var cut lemma -0=0(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subLessRight(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma addEquations(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var fu end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma addEquations(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var fu end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer 1rule adhoc eqAddition(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqAdditionLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fu end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fu end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma distributionOut(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma distributionOut(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed lemma distribution(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma x*0+x=x(R) as system Q infer all metavar var fx end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma x*0+x=x(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed lemma times1(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqAdditionLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma distribution(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plus0Left(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqMultiplicationLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity5 modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "

--------(24.10.06)




" [ math define statement of lemma x*0=0(R)fff as system Q infer all metavar var fx end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma x*0=0(R)fff as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed lemma x=x+(y-y)(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plusAssociativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma x*0+x=x(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule adhoc eqAddition(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma negative(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity5 modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma times(-1)(R) as system Q infer all metavar var fx end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma times(-1)(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed lemma negative(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plusCommutativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqMultiplicationLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma x*0=0(R)fff conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma distribution(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma positiveToRight(Eq)(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plus0Left(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma times1(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqNegated(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma times(-1)Left(R) as system Q infer all metavar var fx end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma times(-1)Left(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed lemma times(-1)(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma timesCommutativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "



(************)

" [ math define statement of lemma -x-y=-(x+y)(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "


" [ math define proof of lemma -x-y=-(x+y)(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed lemma times(-1)Left(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma times(-1)Left(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma addEquations(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma distributionOut(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma times(-1)Left(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity4 modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma lessTotality(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var end define end math ] "

" [ math define proof of lemma lessTotality(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma fromNotLess(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma !!==Symmetry modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma leqNeqLess(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var cut 1rule repetition modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma sameNumerical(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma sameNumerical(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma nonnegativeNumerical(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subLeqRight(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma nonnegativeNumerical(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity4 modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma toLess(R) modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var cut lemma negativeNumerical(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subLessLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var cut lemma negativeNumerical(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqNegated(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity4 modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma from negations modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule mp modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fy end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma minusNegated(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma minusNegated(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed lemma doubleMinus(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule adhoc eqAddition(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma -x-y=-(x+y)(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma plusCommutativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqNegated(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity4 modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "


-----------(24.10.06)

" [ math define statement of lemma positiveNumerical(R) as system Q infer all metavar var fx end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma positiveNumerical(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var infer lemma lessLeq(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma nonnegativeNumerical(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma signNumerical(+)(R) as system Q infer all metavar var fx end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "


" [ math define proof of lemma signNumerical(+)(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var infer lemma positiveNumerical(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma positiveNegated(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var cut lemma negativeNumerical(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma doubleMinus(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma signNumerical(R) as system Q infer all metavar var fx end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma signNumerical(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var infer lemma negativeNegated(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var cut lemma signNumerical(+)(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma doubleMinus(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma sameNumerical(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma eqNegated(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma -0=0(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity4 modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma sameNumerical(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var infer lemma signNumerical(+)(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule deduction modus ponens all metavar var fx end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule deduction modus ponens all metavar var fx end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma lessTotality(R) conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var cut prop lemma from three disjuncts modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma numericalDifference(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma numericalDifference(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed lemma signNumerical(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma minusNegated(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma sameNumerical(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] + [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma - [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair end set | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "

--------(25.10.06)

" [ math define statement of lemma x<=|x|(R) as system Q infer all metavar var fx end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "


" [ math define proof of lemma x<=|x|(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma nonnegativeNumerical(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqLeq(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma 0<=|x|(R) conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma leqTransitivity(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule deduction modus ponens all metavar var fx end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma from leqGeq(R) modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ |f metavar var fx end metavar | ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ |f metavar var fx end metavar | ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "


" [ math var a end math ] "

venter----


\appendix

" [ flush left math define priority of sup2 as preassociative priority sup2 equal priority base equal priority bracket x end bracket equal priority big bracket x end bracket equal priority math x end math equal priority flush left x end left equal priority var x equal priority var y equal priority var z equal priority proclaim x as x end proclaim equal priority define x of x as x end define equal priority pyk equal priority tex equal priority tex name equal priority priority equal priority x equal priority true equal priority if x then x else x end if equal priority introduce x of x as x end introduce equal priority value equal priority claim equal priority bottom equal priority function f of x end function equal priority identity x end identity equal priority false equal priority untagged zero equal priority untagged one equal priority untagged two equal priority untagged three equal priority untagged four equal priority untagged five equal priority untagged six equal priority untagged seven equal priority untagged eight equal priority untagged nine equal priority zero equal priority one equal priority two equal priority three equal priority four equal priority five equal priority six equal priority seven equal priority eight equal priority nine equal priority var a equal priority var b equal priority var c equal priority var d equal priority var e equal priority var f equal priority var g equal priority var h equal priority var i equal priority var j equal priority var k equal priority var l equal priority var m equal priority var n equal priority var o equal priority var p equal priority var q equal priority var r equal priority var s equal priority var t equal priority var u equal priority var v equal priority var w equal priority tagged parenthesis x end tagged equal priority tagged if x then x else x end if equal priority array x is x end array equal priority left equal priority center equal priority right equal priority empty equal priority substitute x set x to x end substitute equal priority map tag x end tag equal priority raw map untag x end untag equal priority map untag x end untag equal priority normalizing untag x end untag equal priority apply x to x end apply equal priority apply one x to x end apply equal priority identifier x end identifier equal priority identifier one x plus id x end identifier equal priority array plus x and x end plus equal priority array remove x array x level x end remove equal priority array put x value x array x level x end put equal priority array add x value x index x value x level x end add equal priority bit x of x end bit equal priority bit one x of x end bit equal priority example rack equal priority vector hook equal priority bibliography hook equal priority dictionary hook equal priority body hook equal priority codex hook equal priority expansion hook equal priority code hook equal priority cache hook equal priority diagnose hook equal priority pyk aspect equal priority tex aspect equal priority texname aspect equal priority value aspect equal priority message aspect equal priority macro aspect equal priority definition aspect equal priority unpack aspect equal priority claim aspect equal priority priority aspect equal priority lambda identifier equal priority apply identifier equal priority true identifier equal priority if identifier equal priority quote identifier equal priority proclaim identifier equal priority define identifier equal priority introduce identifier equal priority hide identifier equal priority pre identifier equal priority post identifier equal priority eval x stack x cache x end eval equal priority eval two x ref x id x stack x cache x end eval equal priority eval three x function x stack x cache x end eval equal priority eval four x arguments x stack x cache x end eval equal priority lookup x stack x default x end lookup equal priority abstract x term x stack x cache x end abstract equal priority quote x end quote equal priority expand x state x cache x end expand equal priority expand two x definition x state x cache x end expand equal priority expand list x state x cache x end expand equal priority macro equal priority macro state equal priority zip x with x end zip equal priority assoc one x address x index x end assoc equal priority protect x end protect equal priority self equal priority macro define x as x end define equal priority value define x as x end define equal priority intro define x as x end define equal priority pyk define x as x end define equal priority tex define x as x end define equal priority tex name define x as x end define equal priority priority table x end table equal priority macro define one equal priority macro define two x end define equal priority macro define three x end define equal priority macro define four x state x cache x definition x end define equal priority state expand x state x cache x end expand equal priority quote expand x term x stack x end expand equal priority quote expand two x term x stack x end expand equal priority quote expand three x term x stack x value x end expand equal priority quote expand star x term x stack x end expand equal priority parenthesis x end parenthesis equal priority big parenthesis x end parenthesis equal priority display x end display equal priority statement x end statement equal priority spying test x end test equal priority false spying test x end test equal priority aspect x subcodex x end aspect equal priority aspect x term x cache x end aspect equal priority tuple x end tuple equal priority tuple one x end tuple equal priority tuple two x end tuple equal priority let two x apply x end let equal priority let one x apply x end let equal priority claim define x as x end define equal priority checker equal priority check x cache x end check equal priority check two x cache x def x end check equal priority check three x cache x def x end check equal priority check list x cache x end check equal priority check list two x cache x value x end check equal priority test x end test equal priority false test x end test equal priority raw test x end test equal priority message equal priority message define x as x end define equal priority the statement aspect equal priority statement equal priority statement define x as x end define equal priority example axiom equal priority example scheme equal priority example rule equal priority absurdity equal priority contraexample equal priority example theory primed equal priority example lemma equal priority metavar x end metavar equal priority meta a equal priority meta b equal priority meta c equal priority meta d equal priority meta e equal priority meta f equal priority meta g equal priority meta h equal priority meta i equal priority meta j equal priority meta k equal priority meta l equal priority meta m equal priority meta n equal priority meta o equal priority meta p equal priority meta q equal priority meta r equal priority meta s equal priority meta t equal priority meta u equal priority meta v equal priority meta w equal priority meta x equal priority meta y equal priority meta z equal priority sub x set x to x end sub equal priority sub star x set x to x end sub equal priority the empty set equal priority example remainder equal priority make visible x end visible equal priority intro x index x pyk x tex x end intro equal priority intro x pyk x tex x end intro equal priority error x term x end error equal priority error two x term x end error equal priority proof x term x cache x end proof equal priority proof two x term x end proof equal priority sequent eval x term x end eval equal priority seqeval init x term x end eval equal priority seqeval modus x term x end eval equal priority seqeval modus one x term x sequent x end eval equal priority seqeval verify x term x end eval equal priority seqeval verify one x term x sequent x end eval equal priority sequent eval plus x term x end eval equal priority seqeval plus one x term x sequent x end eval equal priority seqeval minus x term x end eval equal priority seqeval minus one x term x sequent x end eval equal priority seqeval deref x term x end eval equal priority seqeval deref one x term x sequent x end eval equal priority seqeval deref two x term x sequent x def x end eval equal priority seqeval at x term x end eval equal priority seqeval at one x term x sequent x end eval equal priority seqeval infer x term x end eval equal priority seqeval infer one x term x premise x sequent x end eval equal priority seqeval endorse x term x end eval equal priority seqeval endorse one x term x side x sequent x end eval equal priority seqeval est x term x end eval equal priority seqeval est one x term x name x sequent x end eval equal priority seqeval est two x term x name x sequent x def x end eval equal priority seqeval all x term x end eval equal priority seqeval all one x term x variable x sequent x end eval equal priority seqeval cut x term x end eval equal priority seqeval cut one x term x forerunner x end eval equal priority seqeval cut two x term x forerunner x sequent x end eval equal priority computably true x end true equal priority claims x cache x ref x end claims equal priority claims two x cache x ref x end claims equal priority the proof aspect equal priority proof equal priority lemma x says x end lemma equal priority proof of x reads x end proof equal priority in theory x lemma x says x end lemma equal priority in theory x antilemma x says x end antilemma equal priority in theory x rule x says x end rule equal priority in theory x antirule x says x end antirule equal priority verifier equal priority verify one x end verify equal priority verify two x proofs x end verify equal priority verify three x ref x sequents x diagnose x end verify equal priority verify four x premises x end verify equal priority verify five x ref x array x sequents x end verify equal priority verify six x ref x list x sequents x end verify equal priority verify seven x ref x id x sequents x end verify equal priority cut x and x end cut equal priority head x end head equal priority tail x end tail equal priority rule one x theory x end rule equal priority rule x subcodex x end rule equal priority rule tactic equal priority plus x and x end plus equal priority theory x end theory equal priority theory two x cache x end theory equal priority theory three x name x end theory equal priority theory four x name x sum x end theory equal priority example axiom lemma primed equal priority example scheme lemma primed equal priority example rule lemma primed equal priority contraexample lemma primed equal priority example axiom lemma equal priority example scheme lemma equal priority example rule lemma equal priority contraexample lemma equal priority example theory equal priority ragged right equal priority ragged right expansion equal priority parameter term x stack x seed x end parameter equal priority parameter term star x stack x seed x end parameter equal priority instantiate x with x end instantiate equal priority instantiate star x with x end instantiate equal priority occur x in x substitution x end occur equal priority occur star x in x substitution x end occur equal priority unify x with x substitution x end unify equal priority unify star x with x substitution x end unify equal priority unify two x with x substitution x end unify equal priority ell a equal priority ell b equal priority ell c equal priority ell d equal priority ell e equal priority ell f equal priority ell g equal priority ell h equal priority ell i equal priority ell j equal priority ell k equal priority ell l equal priority ell m equal priority ell n equal priority ell o equal priority ell p equal priority ell q equal priority ell r equal priority ell s equal priority ell t equal priority ell u equal priority ell v equal priority ell w equal priority ell x equal priority ell y equal priority ell z equal priority ell big a equal priority ell big b equal priority ell big c equal priority ell big d equal priority ell big e equal priority ell big f equal priority ell big g equal priority ell big h equal priority ell big i equal priority ell big j equal priority ell big k equal priority ell big l equal priority ell big m equal priority ell big n equal priority ell big o equal priority ell big p equal priority ell big q equal priority ell big r equal priority ell big s equal priority ell big t equal priority ell big u equal priority ell big v equal priority ell big w equal priority ell big x equal priority ell big y equal priority ell big z equal priority ell dummy equal priority sequent reflexivity equal priority tactic reflexivity equal priority sequent commutativity equal priority tactic commutativity equal priority the tactic aspect equal priority tactic equal priority tactic define x as x end define equal priority proof expand x state x cache x end expand equal priority proof expand list x state x cache x end expand equal priority proof state equal priority conclude one x cache x end conclude equal priority conclude two x proves x cache x end conclude equal priority conclude three x proves x lemma x substitution x end conclude equal priority conclude four x lemma x end conclude equal priority check equal priority general macro define x as x end define equal priority make root visible x end visible equal priority sequent example axiom equal priority sequent example rule equal priority sequent example contradiction equal priority sequent example theory equal priority sequent example lemma equal priority set x end set equal priority object var x end var equal priority object a equal priority object b equal priority object c equal priority object d equal priority object e equal priority object f equal priority object g equal priority object h equal priority object i equal priority object j equal priority object k equal priority object l equal priority object m equal priority object n equal priority object o equal priority object p equal priority object q equal priority object r equal priority object s equal priority object t equal priority object u equal priority object v equal priority object w equal priority object x equal priority object y equal priority object z equal priority sub x is x where x is x end sub equal priority sub zero x is x where x is x end sub equal priority sub one x is x where x is x end sub equal priority sub star x is x where x is x end sub equal priority deduction x conclude x end deduction equal priority deduction zero x conclude x end deduction equal priority deduction one x conclude x condition x end deduction equal priority deduction two x conclude x condition x end deduction equal priority deduction three x conclude x condition x bound x end deduction equal priority deduction four x conclude x condition x bound x end deduction equal priority deduction four star x conclude x condition x bound x end deduction equal priority deduction five x condition x bound x end deduction equal priority deduction six x conclude x exception x bound x end deduction equal priority deduction six star x conclude x exception x bound x end deduction equal priority deduction seven x end deduction equal priority deduction eight x bound x end deduction equal priority deduction eight star x bound x end deduction equal priority system s equal priority double negation equal priority rule mp equal priority rule gen equal priority deduction equal priority axiom s one equal priority axiom s two equal priority axiom s three equal priority axiom s four equal priority axiom s five equal priority axiom s six equal priority axiom s seven equal priority axiom s eight equal priority axiom s nine equal priority repetition equal priority lemma a one equal priority lemma a two equal priority lemma a four equal priority lemma a five equal priority prop three two a equal priority prop three two b equal priority prop three two c equal priority prop three two d equal priority prop three two e one equal priority prop three two e two equal priority prop three two e equal priority prop three two f one equal priority prop three two f two equal priority prop three two f equal priority prop three two g one equal priority prop three two g two equal priority prop three two g equal priority prop three two h one equal priority prop three two h two equal priority prop three two h equal priority block one x state x cache x end block equal priority block two x end block equal priority kvanti equal priority lemma uniqueMember equal priority lemma uniqueMember(Type) equal priority lemma sameSeries equal priority lemma a4 equal priority lemma sameMember equal priority 1rule Qclosed(Addition) equal priority 1rule Qclosed(Multiplication) equal priority 1rule fromCartProd(1) equal priority 1rule fromCartProd(2) equal priority constantRationalSeries( x ) equal priority cartProd( x , x ) equal priority P( x ) equal priority binaryUnion( x , x ) equal priority setOfRationalSeries equal priority isSubset( x , x ) equal priority (p x , x ) equal priority (s x ) equal priority cdots equal priority object-var equal priority ex-var equal priority ph-var equal priority vaerdi equal priority variabel equal priority op x end op equal priority op2 x comma x end op2 equal priority define-equal x comma x end equal equal priority contains-empty x end empty equal priority Nat( x ) equal priority 1deduction x conclude x end 1deduction equal priority 1deduction zero x conclude x end 1deduction equal priority 1deduction side x conclude x condition x end 1deduction equal priority 1deduction one x conclude x condition x end 1deduction equal priority 1deduction two x conclude x condition x end 1deduction equal priority 1deduction three x conclude x condition x bound x end 1deduction equal priority 1deduction four x conclude x condition x bound x end 1deduction equal priority 1deduction four star x conclude x condition x bound x end 1deduction equal priority 1deduction five x condition x bound x end 1deduction equal priority 1deduction six x conclude x exception x bound x end 1deduction equal priority 1deduction six star x conclude x exception x bound x end 1deduction equal priority 1deduction seven x end 1deduction equal priority 1deduction eight x bound x end 1deduction equal priority 1deduction eight star x bound x end 1deduction equal priority ex1 equal priority ex2 equal priority ex3 equal priority ex10 equal priority ex20 equal priority existential var x end var equal priority x is existential var equal priority exist-sub x is x where x is x end sub equal priority exist-sub0 x is x where x is x end sub equal priority exist-sub1 x is x where x is x end sub equal priority exist-sub* x is x where x is x end sub equal priority ph1 equal priority ph2 equal priority ph3 equal priority placeholder-var x end var equal priority x is placeholder-var equal priority ph-sub x is x where x is x end sub equal priority ph-sub0 x is x where x is x end sub equal priority ph-sub1 x is x where x is x end sub equal priority ph-sub* x is x where x is x end sub equal priority meta-sub x is x where x is x end sub equal priority meta-sub1 x is x where x is x end sub equal priority meta-sub* x is x where x is x end sub equal priority var big set equal priority object big set equal priority meta big set equal priority zermelo empty set equal priority system Q equal priority 1rule mp equal priority 1rule gen equal priority 1rule repetition equal priority 1rule ad absurdum equal priority 1rule deduction equal priority 1rule exist intro equal priority axiom extensionality equal priority axiom empty set equal priority axiom pair definition equal priority axiom union definition equal priority axiom power definition equal priority axiom separation definition equal priority prop lemma add double neg equal priority prop lemma remove double neg equal priority prop lemma and commutativity equal priority prop lemma auto imply equal priority prop lemma contrapositive equal priority prop lemma first conjunct equal priority prop lemma second conjunct equal priority prop lemma from contradiction equal priority prop lemma from disjuncts equal priority prop lemma iff commutativity equal priority prop lemma iff first equal priority prop lemma iff second equal priority prop lemma imply transitivity equal priority prop lemma join conjuncts equal priority prop lemma mp2 equal priority prop lemma mp3 equal priority prop lemma mp4 equal priority prop lemma mp5 equal priority prop lemma mt equal priority prop lemma negative mt equal priority prop lemma technicality equal priority prop lemma weakening equal priority prop lemma weaken or first equal priority prop lemma weaken or second equal priority lemma formula2pair equal priority lemma pair2formula equal priority lemma formula2union equal priority lemma union2formula equal priority lemma formula2separation equal priority lemma separation2formula equal priority lemma formula2power equal priority lemma subset in power set equal priority lemma power set is subset0 equal priority lemma power set is subset equal priority lemma power set is subset0-switch equal priority lemma power set is subset-switch equal priority lemma set equality suff condition equal priority lemma set equality suff condition(t)0 equal priority lemma set equality suff condition(t) equal priority lemma set equality skip quantifier equal priority lemma set equality nec condition equal priority lemma reflexivity0 equal priority lemma reflexivity equal priority lemma symmetry0 equal priority lemma symmetry equal priority lemma transitivity0 equal priority lemma transitivity equal priority lemma er is reflexive equal priority lemma er is symmetric equal priority lemma er is transitive equal priority lemma empty set is subset equal priority lemma member not empty0 equal priority lemma member not empty equal priority lemma unique empty set0 equal priority lemma unique empty set equal priority lemma ==Reflexivity equal priority lemma ==Symmetry equal priority lemma ==Transitivity0 equal priority lemma ==Transitivity equal priority lemma transfer ~is0 equal priority lemma transfer ~is equal priority lemma pair subset0 equal priority lemma pair subset1 equal priority lemma pair subset equal priority lemma same pair equal priority lemma same singleton equal priority lemma union subset equal priority lemma same union equal priority lemma separation subset equal priority lemma same separation equal priority lemma same binary union equal priority lemma intersection subset equal priority lemma same intersection equal priority lemma auto member equal priority lemma eq-system not empty0 equal priority lemma eq-system not empty equal priority lemma eq subset0 equal priority lemma eq subset equal priority lemma equivalence nec condition0 equal priority lemma equivalence nec condition equal priority lemma none-equivalence nec condition0 equal priority lemma none-equivalence nec condition1 equal priority lemma none-equivalence nec condition equal priority lemma equivalence class is subset equal priority lemma equivalence classes are disjoint equal priority lemma all disjoint equal priority lemma all disjoint-imply equal priority lemma bs subset union(bs/r) equal priority lemma union(bs/r) subset bs equal priority lemma union(bs/r) is bs equal priority theorem eq-system is partition equal priority var x1 equal priority var x2 equal priority var y1 equal priority var y2 equal priority var v1 equal priority var v2 equal priority var v3 equal priority var v4 equal priority var v2n equal priority var m1 equal priority var m2 equal priority var n1 equal priority var n2 equal priority var n3 equal priority var ep equal priority var ep1 equal priority var ep2 equal priority var fep equal priority var fx equal priority var fy equal priority var fz equal priority var fu equal priority var fv equal priority var fw equal priority var rx equal priority var ry equal priority var rz equal priority var ru equal priority var sx equal priority var sx1 equal priority var sy equal priority var sy1 equal priority var sz equal priority var sz1 equal priority var su equal priority var su1 equal priority var fxs equal priority var fys equal priority var crs1 equal priority var f1 equal priority var f2 equal priority var f3 equal priority var f4 equal priority var op1 equal priority var op2 equal priority var r1 equal priority var s1 equal priority var s2 equal priority meta x1 equal priority meta x2 equal priority meta y1 equal priority meta y2 equal priority meta v1 equal priority meta v2 equal priority meta v3 equal priority meta v4 equal priority meta v2n equal priority meta m1 equal priority meta m2 equal priority meta n1 equal priority meta n2 equal priority meta n3 equal priority meta ep equal priority meta ep1 equal priority meta ep2 equal priority meta fx equal priority meta fy equal priority meta fz equal priority meta fu equal priority meta fv equal priority meta fw equal priority meta fep equal priority meta rx equal priority meta ry equal priority meta rz equal priority meta ru equal priority meta sx equal priority meta sx1 equal priority meta sy equal priority meta sy1 equal priority meta sz equal priority meta sz1 equal priority meta su equal priority meta su1 equal priority meta fxs equal priority meta fys equal priority meta f1 equal priority meta f2 equal priority meta f3 equal priority meta f4 equal priority meta op1 equal priority meta op2 equal priority meta r1 equal priority meta s1 equal priority meta s2 equal priority object ep equal priority object crs1 equal priority object f1 equal priority object f2 equal priority object f3 equal priority object f4 equal priority object n1 equal priority object n2 equal priority object op1 equal priority object op2 equal priority object r1 equal priority object s1 equal priority object s2 equal priority ph4 equal priority ph5 equal priority ph6 equal priority NAT equal priority RATIONAL_SERIES equal priority SERIES equal priority setOfReals equal priority setOfFxs equal priority N equal priority Q equal priority X equal priority xs equal priority xsF equal priority ysF equal priority us equal priority usF equal priority 0 equal priority 1 equal priority (-1) equal priority 2 equal priority 3 equal priority 1/2 equal priority 1/3 equal priority 2/3 equal priority 0f equal priority 1f equal priority 00 equal priority 01 equal priority (--01) equal priority 02 equal priority 01//02 equal priority lemma plusAssociativity(R) equal priority lemma plusAssociativity(R)XX equal priority lemma plus0(R) equal priority lemma negative(R) equal priority lemma times1(R) equal priority lemma lessAddition(R) equal priority lemma plusCommutativity(R) equal priority lemma leqAntisymmetry(R) equal priority lemma leqTransitivity(R) equal priority lemma leqAddition(R) equal priority lemma distribution(R) equal priority axiom a4 equal priority axiom induction equal priority axiom equality equal priority axiom eqLeq equal priority axiom eqAddition equal priority axiom eqMultiplication equal priority axiom QisClosed(reciprocal) equal priority lemma QisClosed(reciprocal) equal priority axiom QisClosed(negative) equal priority lemma QisClosed(negative) equal priority axiom leqReflexivity equal priority axiom leqAntisymmetry equal priority axiom leqTransitivity equal priority axiom leqTotality equal priority axiom leqAddition equal priority axiom leqMultiplication equal priority axiom plusAssociativity equal priority axiom plusCommutativity equal priority axiom negative equal priority axiom plus0 equal priority axiom timesAssociativity equal priority axiom timesCommutativity equal priority axiom reciprocal equal priority axiom times1 equal priority axiom distribution equal priority axiom 0not1 equal priority lemma eqLeq(R) equal priority lemma timesAssociativity(R) equal priority lemma timesCommutativity(R) equal priority 1rule adhoc sameR equal priority lemma separation2formula(1) equal priority lemma separation2formula(2) equal priority axiom cauchy equal priority axiom plusF equal priority axiom reciprocalF equal priority 1rule from== equal priority 1rule to== equal priority 1rule fromInR equal priority lemma plusR(Sym) equal priority axiom reciprocalR equal priority 1rule lessMinus1(N) equal priority axiom nonnegative(N) equal priority axiom US0 equal priority 1rule nextXS(upperBound) equal priority 1rule nextXS(noUpperBound) equal priority 1rule nextUS(upperBound) equal priority 1rule nextUS(noUpperBound) equal priority 1rule expZero equal priority 1rule expPositive equal priority 1rule expZero(R) equal priority 1rule expPositive(R) equal priority 1rule base(1/2)Sum zero equal priority 1rule base(1/2)Sum positive equal priority 1rule UStelescope zero equal priority 1rule UStelescope positive equal priority 1rule adhoc eqAddition(R) equal priority 1rule fromLimit equal priority 1rule toUpperBound equal priority 1rule fromUpperBound equal priority axiom USisUpperBound equal priority axiom 0not1(R) equal priority 1rule expUnbounded equal priority 1rule fromLeq(Advanced)(N) equal priority 1rule fromLeastUpperBound equal priority 1rule toLeastUpperBound equal priority axiom XSisNotUpperBound equal priority axiom ysFGreater equal priority axiom ysFLess equal priority 1rule smallInverse equal priority axiom natType equal priority axiom rationalType equal priority axiom seriesType equal priority axiom max equal priority axiom numerical equal priority axiom numericalF equal priority axiom memberOfSeries equal priority prop lemma doubly conditioned join conjuncts equal priority prop lemma imply negation equal priority prop lemma tertium non datur equal priority prop lemma from negated imply equal priority prop lemma to negated imply equal priority prop lemma from negated double imply equal priority prop lemma from negated and equal priority prop lemma from negated or equal priority prop lemma to negated or equal priority prop lemma from negations equal priority prop lemma from three disjuncts equal priority prop lemma from two times two disjuncts equal priority prop lemma negate first disjunct equal priority prop lemma negate second disjunct equal priority prop lemma expand disjuncts equal priority lemma set equality nec condition(1) equal priority lemma set equality nec condition(2) equal priority lemma lessLeq(R) equal priority lemma memberOfSeries equal priority lemma memberOfSeries(Type) equal priority sup equal priority prop lemma to negated and(1) equal priority lemma uniqueNegative equal priority lemma doubleMinus equal priority lemma minusNegated equal priority lemma eqReflexivity equal priority lemma eqSymmetry equal priority lemma eqTransitivity equal priority lemma eqTransitivity4 equal priority lemma eqTransitivity5 equal priority lemma eqTransitivity6 equal priority lemma addEquations equal priority lemma subtractEquations equal priority lemma subtractEquationsLeft equal priority lemma multiplyEquations equal priority lemma eqNegated equal priority lemma positiveToRight(Eq) equal priority lemma positiveToLeft(Eq)(1 term) equal priority lemma negativeToLeft(Eq) equal priority lemma nonreciprocalToRight(Eq)(1 term) equal priority lemma plusAssociativity(4 terms) equal priority lemma lessNeq equal priority lemma neqSymmetry equal priority lemma neqNegated equal priority lemma subNeqRight equal priority lemma subNeqLeft equal priority lemma negativeToRight(Neq)(1 term) equal priority lemma neqAddition equal priority lemma neqMultiplication equal priority lemma nonzeroProduct(2) equal priority lemma UStelescope(+1) equal priority lemma telescopeBound base equal priority lemma telescopeBound indu equal priority lemma telescopeBound equal priority lemma intervalSize base equal priority lemma intervalSize indu equal priority lemma intervalSize equal priority lemma XSlessUS equal priority lemma USdecreasing(+1) equal priority lemma closeUS equal priority lemma closeUS(n+1) equal priority pred lemma allNegated(Imply) equal priority pred lemma existNegated(Imply) equal priority pred lemma intro exist helper equal priority pred lemma intro exist equal priority pred lemma exist mp equal priority pred lemma exist mp2 equal priority pred lemma 2exist mp equal priority pred lemma 2exist mp2 equal priority pred lemma EAE mp equal priority pred lemma addAll equal priority pred lemma addExist helper1 equal priority pred lemma addExist helper2 equal priority pred lemma addExist equal priority pred lemma addExist(SimpleAnt) equal priority pred lemma addExist(Simple) equal priority pred lemma addEAE equal priority pred lemma AEAnegated equal priority pred lemma EEAnegated equal priority lemma induction equal priority lemma leqAntisymmetry equal priority lemma leqTransitivity equal priority lemma leqAddition equal priority lemma leqMultiplication equal priority lemma reciprocal equal priority lemma equality equal priority lemma eqLeq equal priority lemma eqAddition equal priority lemma eqMultiplication equal priority lemma leqMultiplicationLeft equal priority lemma leqLessEq equal priority lemma lessLeq equal priority lemma from leqGeq equal priority lemma subLeqRight equal priority lemma subLeqLeft equal priority lemma leqPlus1 equal priority lemma positiveToRight(Leq) equal priority lemma positiveToRight(Leq)(1 term) equal priority lemma negativeToRight(Leq) equal priority lemma positiveToLeft(Leq) equal priority lemma negativeToLeft(Leq) equal priority lemma negativeToLeft(Leq)(1 term) equal priority lemma leqAdditionLeft equal priority lemma leqSubtraction equal priority lemma leqSubtractionLeft equal priority lemma thirdGeq equal priority lemma leqNegated equal priority lemma addEquations(Leq) equal priority lemma multiplyEquations(Leq) equal priority lemma thirdGeqSeries equal priority lemma leqNeqLess equal priority lemma fromLess equal priority lemma toLess equal priority lemma fromNotLess equal priority lemma toNotLess equal priority lemma negativeLessPositive equal priority lemma leqLessTransitivity equal priority lemma lessLeqTransitivity equal priority lemma lessTransitivity equal priority lemma lessTotality equal priority lemma subLessRight equal priority lemma subLessLeft equal priority lemma switchTerms(x
\section{Pyk definitioner} \label{sec:pyk}

\begin{flushleft}
" [ math define pyk of lemma leqTotality(R) as text "lemma leqTotality(R)" end text end define linebreak define pyk of lemma positiveToLeft(Eq) as text "lemma positiveToLeft(Eq)" end text end define linebreak define pyk of lemma expZero exact as text "lemma expZero exact" end text end define linebreak define pyk of lemma sameExp base as text "lemma sameExp base" end text end define linebreak define pyk of lemma sameExp indu as text "lemma sameExp indu" end text end define linebreak define pyk of lemma sameExp as text "lemma sameExp" end text end define linebreak define pyk of lemma (1/2)(x+y)-x=(1/2)(y-x) as text "lemma (1/2)(x+y)-x=(1/2)(y-x)" end text end define linebreak define pyk of lemma y-(1/2)(x+y)=(1/2)(y-x) as text "lemma y-(1/2)(x+y)=(1/2)(y-x)" end text end define linebreak define pyk of lemma base(1/2)Sum zero exact as text "lemma base(1/2)Sum zero exact" end text end define linebreak define pyk of lemma sameBase(1/2)Sum second base as text "lemma sameBase(1/2)Sum second base" end text end define linebreak define pyk of lemma sameBase(1/2)Sum second indu as text "lemma sameBase(1/2)Sum second indu" end text end define linebreak define pyk of lemma sameBase(1/2)Sum second as text "lemma sameBase(1/2)Sum second" end text end define linebreak define pyk of lemma negativeToLeft(Less)(1 term) as text "lemma negativeToLeft(Less)(1 term)" end text end define linebreak define pyk of lemma base(1/2)Sum(+1) as text "lemma base(1/2)Sum(+1)" end text end define linebreak define pyk of lemma base(1/2)Sum exact bound base as text "lemma base(1/2)Sum exact bound base" end text end define linebreak define pyk of lemma base(1/2)Sum exact bound indu as text "lemma base(1/2)Sum exact bound indu" end text end define linebreak define pyk of lemma base(1/2)Sum exact bound as text "lemma base(1/2)Sum exact bound" end text end define linebreak define pyk of lemma base(1/2)Sum bound as text "lemma base(1/2)Sum bound" end text end define linebreak define pyk of lemma UStelescope zero exact as text "lemma UStelescope zero exact" end text end define linebreak define pyk of lemma sameTelescope second base as text "lemma sameTelescope second base" end text end define linebreak define pyk of lemma sameTelescope second indu as text "lemma sameTelescope second indu" end text end define linebreak define pyk of lemma sameTelescope second as text "lemma sameTelescope second" end text end define linebreak define pyk of lemma exp(+1) as text "lemma exp(+1)" end text end define linebreak define pyk of lemma positiveBase base as text "lemma positiveBase base" end text end define linebreak define pyk of lemma positiveBase indu as text "lemma positiveBase indu" end text end define linebreak define pyk of lemma positiveBase as text "lemma positiveBase" end text end define linebreak define pyk of lemma telescopeNumerical base as text "lemma telescopeNumerical base" end text end define linebreak define pyk of lemma telescopeNumerical indu as text "lemma telescopeNumerical indu" end text end define linebreak define pyk of lemma telescopeNumerical as text "lemma telescopeNumerical" end text end define linebreak define pyk of lemma +1IsPositive(N) as text "lemma +1IsPositive(N)" end text end define linebreak define pyk of lemma distributionOut(Minus) as text "lemma distributionOut(Minus)" end text end define linebreak define pyk of lemma positiveToRight(Eq)(1 term) as text "lemma positiveToRight(Eq)(1 term)" end text end define linebreak define pyk of lemma sameSeries(NumDiff) as text "lemma sameSeries(NumDiff)" end text end define linebreak define pyk of prop lemma to negated double imply as text "prop lemma to negated double imply" end text end define linebreak define pyk of pred lemma addNegatedAll as text "pred lemma addNegatedAll" end text end define linebreak define pyk of pred lemma (A)to(~E~)(Imply) as text "pred lemma (A)to(~E~)(Imply)" end text end define linebreak define pyk of pred lemma (E)to(~A~)(Imply) as text "pred lemma (E)to(~A~)(Imply)" end text end define linebreak define pyk of pred lemma (E~)to(~A)(Imply) as text "pred lemma (E~)to(~A)(Imply)" end text end define linebreak define pyk of pred lemma toNegatedAEA as text "pred lemma toNegatedAEA" end text end define linebreak define pyk of lemma three2threeTerms(R) as text "lemma three2threeTerms(R)" end text end define linebreak define pyk of lemma lessNeq(F) helper as text "lemma lessNeq(F) helper" end text end define linebreak define pyk of lemma lessNeq(F) as text "lemma lessNeq(F)" end text end define linebreak define pyk of lemma lessNeq(R) as text "lemma lessNeq(R)" end text end define linebreak define pyk of lemma x=x+(y-y)(R) as text "lemma x=x+(y-y)(R)" end text end define linebreak define pyk of lemma x=x+y-y(R) as text "lemma x=x+y-y(R)" end text end define linebreak define pyk of lemma subtractEquations(R) as text "lemma subtractEquations(R)" end text end define linebreak define pyk of lemma neqAddition(R) as text "lemma neqAddition(R)" end text end define linebreak define pyk of lemma positiveToRight(Less)(R) as text "lemma positiveToRight(Less)(R)" end text end define linebreak define pyk of lemma positiveToRight(Less)(1 term)(R) as text "lemma positiveToRight(Less)(1 term)(R)" end text end define linebreak define pyk of lemma leqNeqLess(R) as text "lemma leqNeqLess(R)" end text end define linebreak define pyk of lemma subLeqLeft(R) as text "lemma subLeqLeft(R)" end text end define linebreak define pyk of lemma toLeq(Advanced)(R) as text "lemma toLeq(Advanced)(R)" end text end define linebreak define pyk of lemma leqLessTransitivity(R) as text "lemma leqLessTransitivity(R)" end text end define linebreak define pyk of lemma negativeToLeft(Eq)(R) as text "lemma negativeToLeft(Eq)(R)" end text end define linebreak define pyk of lemma negativeToRight(Less)(R) as text "lemma negativeToRight(Less)(R)" end text end define linebreak define pyk of lemma !!==Symmetry as text "lemma !!==Symmetry" end text end define linebreak define pyk of lemma switchTerms(x<=y-z) as text "lemma switchTerms(x<=y-z)" end text end define linebreak define pyk of lemma plus0Left(R) as text "lemma plus0Left(R)" end text end define linebreak define pyk of lemma positiveToRight(Eq)(R) as text "lemma positiveToRight(Eq)(R)" end text end define linebreak define pyk of lemma eqAdditionLeft(R) as text "lemma eqAdditionLeft(R)" end text end define linebreak define pyk of lemma three2twoTerms(R) as text "lemma three2twoTerms(R)" end text end define linebreak define pyk of lemma to!!== as text "lemma to!!==" end text end define linebreak define pyk of lemma positiveToRight(Less)(1 term) as text "lemma positiveToRight(Less)(1 term)" end text end define linebreak define pyk of pred lemma (A~)to(~E) as text "pred lemma (A~)to(~E)" end text end define linebreak define pyk of lemma negativeToRight(Eq)(R) as text "lemma negativeToRight(Eq)(R)" end text end define linebreak define pyk of lemma negativeToRight(Eq)(1 term)(R) as text "lemma negativeToRight(Eq)(1 term)(R)" end text end define linebreak define pyk of lemma doubleMinus(R) as text "lemma doubleMinus(R)" end text end define linebreak define pyk of lemma uniqueNegative(R) as text "lemma uniqueNegative(R)" end text end define linebreak define pyk of lemma subtractEquationsLeft(R) as text "lemma subtractEquationsLeft(R)" end text end define linebreak define pyk of lemma eqNegated(R) as text "lemma eqNegated(R)" end text end define linebreak define pyk of lemma neqNegated(R) as text "lemma neqNegated(R)" end text end define linebreak define pyk of lemma -0=0(R) as text "lemma -0=0(R)" end text end define linebreak define pyk of lemma negativeNegated(R) as text "lemma negativeNegated(R)" end text end define linebreak define pyk of lemma from leqGeq(R) as text "lemma from leqGeq(R)" end text end define linebreak define pyk of lemma 0<=|x|(R) as text "lemma 0<=|x|(R)" end text end define linebreak define pyk of lemma positiveNegated(R) as text "lemma positiveNegated(R)" end text end define linebreak define pyk of lemma addEquations(R) as text "lemma addEquations(R)" end text end define linebreak define pyk of lemma times(-1)(R) as text "lemma times(-1)(R)" end text end define linebreak define pyk of lemma times(-1)Left(R) as text "lemma times(-1)Left(R)" end text end define linebreak define pyk of lemma -x-y=-(x+y)(R) as text "lemma -x-y=-(x+y)(R)" end text end define linebreak define pyk of lemma lessTotality(R) as text "lemma lessTotality(R)" end text end define linebreak define pyk of lemma sameNumerical(R) as text "lemma sameNumerical(R)" end text end define linebreak define pyk of lemma minusNegated(R) as text "lemma minusNegated(R)" end text end define linebreak define pyk of lemma positiveNumerical(R) as text "lemma positiveNumerical(R)" end text end define linebreak define pyk of lemma signNumerical(+)(R) as text "lemma signNumerical(+)(R)" end text end define linebreak define pyk of lemma nonnegativeNumerical(R) as text "lemma nonnegativeNumerical(R)" end text end define linebreak define pyk of lemma negativeNumerical(R) as text "lemma negativeNumerical(R)" end text end define linebreak define pyk of lemma leqNegated(R) as text "lemma leqNegated(R)" end text end define linebreak define pyk of lemma lessNegated(R) as text "lemma lessNegated(R)" end text end define linebreak define pyk of lemma subLeqRight(R) as text "lemma subLeqRight(R)" end text end define linebreak define pyk of lemma fromLess(R) as text "lemma fromLess(R)" end text end define linebreak define pyk of lemma distributionOut(R) as text "lemma distributionOut(R)" end text end define linebreak define pyk of lemma x*0+x=x(R) as text "lemma x*0+x=x(R)" end text end define linebreak define pyk of lemma x*0=0(R)fff as text "lemma x*0=0(R)fff" end text end define linebreak define pyk of lemma signNumerical(R) as text "lemma signNumerical(R)" end text end define linebreak define pyk of lemma numericalDifference(R) as text "lemma numericalDifference(R)" end text end define linebreak define pyk of lemma x<=|x|(R) as text "lemma x<=|x|(R)" end text end define linebreak define pyk of lemma USlimitIsUpperBound helper as text "lemma USlimitIsUpperBound helper" end text end define linebreak define pyk of lemma USlimitIsUpperBound as text "lemma USlimitIsUpperBound" end text end define linebreak define pyk of lemma (-1)*(-1)+(-1)*1=0(R) as text "lemma (-1)*(-1)+(-1)*1=0(R)" end text end define linebreak define pyk of lemma (-1)*(-1)=1(R) as text "lemma (-1)*(-1)=1(R)" end text end define linebreak define pyk of lemma 0<1Helper(R) as text "lemma 0<1Helper(R)" end text end define linebreak define pyk of lemma 0<1(R) as text "lemma 0<1(R)" end text end define linebreak define pyk of lemma expZero exact(R) as text "lemma expZero exact(R)" end text end define linebreak define pyk of lemma positiveBase(R) base as text "lemma positiveBase(R) base" end text end define linebreak define pyk of lemma three2twoFactors(R) as text "lemma three2twoFactors(R)" end text end define linebreak define pyk of lemma x=x*y*(1/y)(R) as text "lemma x=x*y*(1/y)(R)" end text end define linebreak define pyk of lemma neqMultiplication(R) as text "lemma neqMultiplication(R)" end text end define linebreak define pyk of lemma lessTransitivity(R) as text "lemma lessTransitivity(R)" end text end define linebreak define pyk of lemma 0<2(R) as text "lemma 0<2(R)" end text end define linebreak define pyk of lemma sameExp(R) base as text "lemma sameExp(R) base" end text end define linebreak define pyk of lemma sameExp(R) indu as text "lemma sameExp(R) indu" end text end define linebreak define pyk of lemma sameExp(R) as text "lemma sameExp(R)" end text end define linebreak define pyk of lemma subNeqLeft(R) as text "lemma subNeqLeft(R)" end text end define linebreak define pyk of lemma subNeqRight(R) as text "lemma subNeqRight(R)" end text end define linebreak define pyk of lemma nonzeroFactors(R) as text "lemma nonzeroFactors(R)" end text end define linebreak define pyk of lemma nonnegativeFactors(R) as text "lemma nonnegativeFactors(R)" end text end define linebreak define pyk of lemma positiveFactors(R) as text "lemma positiveFactors(R)" end text end define linebreak define pyk of lemma lessDivision(R) as text "lemma lessDivision(R)" end text end define linebreak define pyk of lemma 0<1/2(R) as text "lemma 0<1/2(R)" end text end define linebreak define pyk of lemma positiveToRight(Eq)(1 term)(R) as text "lemma positiveToRight(Eq)(1 term)(R)" end text end define linebreak define pyk of lemma exp(+1)(R) as text "lemma exp(+1)(R)" end text end define linebreak define pyk of lemma positiveBase(R) indu as text "lemma positiveBase(R) indu" end text end define linebreak define pyk of lemma positiveBase(R) as text "lemma positiveBase(R)" end text end define linebreak define pyk of lemma -x*y=-(x*y)(R) as text "lemma -x*y=-(x*y)(R)" end text end define linebreak define pyk of lemma positiveToLeft(Eq)(R) as text "lemma positiveToLeft(Eq)(R)" end text end define linebreak define pyk of lemma times1Left(R) as text "lemma times1Left(R)" end text end define linebreak define pyk of lemma x+x=2*x(R) as text "lemma x+x=2*x(R)" end text end define linebreak define pyk of lemma (1/2)x+(1/2)x=x(R) as text "lemma (1/2)x+(1/2)x=x(R)" end text end define linebreak define pyk of lemma distributionOut(Minus)(R) as text "lemma distributionOut(Minus)(R)" end text end define linebreak define pyk of lemma (1/2)(x+y)-x=(1/2)(y-x)(R) as text "lemma (1/2)(x+y)-x=(1/2)(y-x)(R)" end text end define linebreak define pyk of lemma intervalSize(R) base as text "lemma intervalSize(R) base" end text end define linebreak define pyk of lemma lessMultiplicationLeft(R) as text "lemma lessMultiplicationLeft(R)" end text end define linebreak define pyk of lemma negativeToLeft(Less)(R) as text "lemma negativeToLeft(Less)(R)" end text end define linebreak define pyk of lemma negativeToLeft(Less)(1 term)(R) as text "lemma negativeToLeft(Less)(1 term)(R)" end text end define linebreak define pyk of lemma y-(1/2)(x+y)=(1/2)(y-x)(R) as text "lemma y-(1/2)(x+y)=(1/2)(y-x)(R)" end text end define linebreak define pyk of lemma intervalSize(R) indu as text "lemma intervalSize(R) indu" end text end define linebreak define pyk of lemma intervalSize(R) as text "lemma intervalSize(R)" end text end define linebreak define pyk of lemma XSlessUS(R) as text "lemma XSlessUS(R)" end text end define linebreak define pyk of lemma USdecreasing(+1)(R) as text "lemma USdecreasing(+1)(R)" end text end define linebreak define pyk of lemma expUnbounded base as text "lemma expUnbounded base" end text end define linebreak define pyk of lemma expUnbounded indu as text "lemma expUnbounded indu" end text end define linebreak define pyk of lemma expUnbounded as text "lemma expUnbounded" end text end define linebreak define pyk of lemma 1<=x+1(N) as text "lemma 1<=x+1(N)" end text end define linebreak define pyk of lemma expNonzero base as text "lemma expNonzero base" end text end define linebreak define pyk of lemma expNonzero indu as text "lemma expNonzero indu" end text end define linebreak define pyk of lemma expNonzero as text "lemma expNonzero" end text end define linebreak define pyk of lemma expNonzero(2) as text "lemma expNonzero(2)" end text end define linebreak define pyk of lemma halfBase base as text "lemma halfBase base" end text end define linebreak define pyk of lemma halfBase indu as text "lemma halfBase indu" end text end define linebreak define pyk of lemma multiplyEquations(R) as text "lemma multiplyEquations(R)" end text end define linebreak define pyk of lemma nonreciprocalToRight(Eq)(1 term)(R) as text "lemma nonreciprocalToRight(Eq)(1 term)(R)" end text end define linebreak define pyk of lemma positiveNonzero(R) as text "lemma positiveNonzero(R)" end text end define linebreak define pyk of lemma nonzeroProduct(2)(R) as text "lemma nonzeroProduct(2)(R)" end text end define linebreak define pyk of lemma halfBase as text "lemma halfBase" end text end define linebreak define pyk of lemma three2threeFactors(R) as text "lemma three2threeFactors(R)" end text end define linebreak define pyk of lemma x*y=zBackwards(R) as text "lemma x*y=zBackwards(R)" end text end define linebreak define pyk of lemma positiveInverted(R) as text "lemma positiveInverted(R)" end text end define linebreak define pyk of lemma reciprocalToRight(Less)(R) as text "lemma reciprocalToRight(Less)(R)" end text end define linebreak define pyk of lemma reciprocalToRight(Less)(1 term)(R) as text "lemma reciprocalToRight(Less)(1 term)(R)" end text end define linebreak define pyk of lemma nonreciprocalToLeft(Less)(R) as text "lemma nonreciprocalToLeft(Less)(R)" end text end define linebreak define pyk of lemma 1 \end{flushleft}

\newpage

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item " [ math define tex of sup2 as text "sup2" end text end define end math ] "

\item " [ math define tex of lemma leqTotality(R) as text "LeqTotality(R)" end text end define end math ] "
\item " [ math define tex of lemma positiveToLeft(Eq) as text "PositiveToLeft(Eq)" end text end define end math ] "
\item " [ math define tex of lemma expZero exact as text "ExpZero(Exact) " end text end define end math ] "
\item " [ math define tex of lemma +1IsPositive(N) as text "(+1)IsPositive(N)" end text end define end math ] "
\item " [ math define tex of lemma sameExp base as text "SameExp(Base)" end text end define end math ] "
\item " [ math define tex of lemma sameExp indu as text "SameExp(Indu)" end text end define end math ] "
\item " [ math define tex of lemma sameExp as text "SameExp" end text end define end math ] "
\item " [ math define tex of lemma exp(+1) as text "Exp(+1)" end text end define end math ] "
\item " [ math define tex of lemma distributionOut(Minus) as text "DistributionOut(Minus)" end text end define end math ] "
\item " [ math define tex of lemma (1/2)(x+y)-x=(1/2)(y-x) as text "(1/2)(x+y)-x=(1/2)(y-x)" end text end define end math ] "
\item " [ math define tex of lemma y-(1/2)(x+y)=(1/2)(y-x) as text "y-(1/2)(x+y)=(1/2)(y-x)" end text end define end math ] "
\item " [ math define tex of lemma positiveBase base as text "PositiveBase(Base)" end text end define end math ] "
\item " [ math define tex of lemma positiveBase indu as text "PositiveBase(Indu)" end text end define end math ] "
\item " [ math define tex of lemma positiveBase as text "PositiveBase" end text end define end math ] "
\item " [ math define tex of lemma positiveToRight(Eq)(1 term) as text "PositiveToRight(Eq)(1 term)" end text end define end math ] "
\item " [ math define tex of lemma base(1/2)Sum zero exact as text "BSzero(Exact)" end text end define end math ] "
\item " [ math define tex of lemma sameBase(1/2)Sum second base as text "SameBS(2)(Base)" end text end define end math ] "
\item " [ math define tex of lemma sameBase(1/2)Sum second indu as text "SameBS(2)(Indu)" end text end define end math ] "
\item " [ math define tex of lemma sameBase(1/2)Sum second as text "SameBS(2)" end text end define end math ] "
\item " [ math define tex of lemma negativeToLeft(Less)(1 term) as text "NegativeToLeft(Less)(1 term)" end text end define end math ] "

\item " [ math define tex of lemma base(1/2)Sum(+1) as text "BS(+1)" end text end define end math ] "
\item " [ math define tex of lemma base(1/2)Sum exact bound base as text "BSbound(Exact)(Base)" end text end define end math ] "
\item " [ math define tex of lemma base(1/2)Sum exact bound indu as text "BSbound(Exact)(Indu)" end text end define end math ] "
\item " [ math define tex of lemma base(1/2)Sum exact bound as text "BSbound(Exact)" end text end define end math ] "
\item " [ math define tex of lemma base(1/2)Sum bound as text "BSbound" end text end define end math ] "
\item " [ math define tex of lemma sameSeries(NumDiff) as text "SameSeries(NumDiff)" end text end define end math ] "
\item " [ math define tex of lemma UStelescope zero exact as text "UStelescope(Zero)(Exact)" end text end define end math ] "
\item " [ math define tex of lemma sameTelescope second base as text "SameTelescope(2)(Base)" end text end define end math ] "
\item " [ math define tex of lemma sameTelescope second indu as text "SameTelescope(2)(Indu)" end text end define end math ] "
\item " [ math define tex of lemma sameTelescope second as text "SameTelescope(2)" end text end define end math ] "
\item " [ math define tex of lemma telescopeNumerical base as text "TelescopeNumerical(Base)" end text end define end math ] "
\item " [ math define tex of lemma telescopeNumerical indu as text "TelescopeNumerical(Indu)" end text end define end math ] "
\item " [ math define tex of lemma telescopeNumerical as text "TelescopeNumerical" end text end define end math ] "
\item " [ math define tex of prop lemma to negated double imply as text "ToNegatedDoubleImply" end text end define end math ] "
\item " [ math define tex of lemma eqAdditionLeft(R) as text "EqAdditionLeft(R)" end text end define end math ] "
\item " [ math define tex of lemma x=x+(y-y)(R) as text "x=x+(y-y)(R)" end text end define end math ] "
\item " [ math define tex of lemma x=x+y-y(R) as text "x=x+y-y(R)" end text end define end math ] "
\item " [ math define tex of lemma three2twoTerms(R) as text "Three2twoTerms(R)" end text end define end math ] "
\item " [ math define tex of lemma positiveToRight(Less)(R) as text "PositiveToRight(Less)(R)" end text end define end math ] "
\item " [ math define tex of lemma three2threeTerms(R) as text "Three2threeTerms(R)" end text end define end math ] "
\item " [ math define tex of lemma plus0Left(R) as text "Plus0Left(R)" end text end define end math ] "
\item " [ math define tex of lemma positiveToRight(Eq)(R) as text "PositiveToRight(Eq)(R)" end text end define end math ] "
\item " [ math define tex of lemma subtractEquations(R) as text "SubtractEquations(R)" end text end define end math ] "
\item " [ math define tex of lemma neqAddition(R) as text "NeqAddition(R)" end text end define end math ] "
\item " [ math define tex of lemma positiveToRight(Less)(R) as text "PositiveToRight(Less)(R)" end text end define end math ] "
\item " [ math define tex of lemma positiveToRight(Less)(1 term)(R) as text "PositiveToRight(Less)(1 term)(R)" end text end define end math ] "
\item " [ math define tex of lemma to!!== as text "To!!==" end text end define end math ] "
\item " [ math define tex of lemma switchTerms(x<=y-z) as text "SwitchTerms(x<=y-z)" end text end define end math ] "
\item " [ math define tex of pred lemma (A)to(~E~)(Imply) as text "(A)to(~E~)(Imply)" end text end define end math ] "
\item " [ math define tex of pred lemma (E)to(~A~)(Imply) as text "(E)to(~A~)(Imply)" end text end define end math ] "
\item " [ math define tex of pred lemma (E~)to(~A)(Imply) as text "(E~)to(~A)(Imply)" end text end define end math ] "
\item " [ math define tex of pred lemma addNegatedAll as text "AddNegatedAll" end text end define end math ] "
\item " [ math define tex of pred lemma toNegatedAEA as text "ToNegatedAEA " end text end define end math ] "
\item " [ math define tex of lemma lessNeq(F) helper as text "LessNeq(F)(Helper)" end text end define end math ] "
\item " [ math define tex of lemma lessNeq(F) as text "LessNeq(F)" end text end define end math ] "
\item " [ math define tex of lemma lessNeq(R) as text "LessNeq(R)" end text end define end math ] "
\item " [ math define tex of lemma positiveToRight(Less)(1 term) as text "PositiveToRight(Less)(1 term)" end text end define end math ] "
\item " [ math define tex of pred lemma (A~)to(~E) as text "(A~)to(~E)" end text end define end math ] "
\item " [ math define tex of lemma toLeq(Advanced)(R) as text "ToLeq(Advanced)(R)" end text end define end math ] "
\item " [ math define tex of lemma leqNeqLess(R) as text "LeqNeqLess(R)" end text end define end math ] "
\item " [ math define tex of lemma subLeqLeft(R) as text "SubLeqLeft(R)" end text end define end math ] "
\item " [ math define tex of lemma leqLessTransitivity(R) as text "LeqLessTransitivity(R)" end text end define end math ] "
\item " [ math define tex of lemma negativeToLeft(Eq)(R) as text "NegativeToLeft(Eq)(R)" end text end define end math ] "
\item " [ math define tex of lemma negativeToRight(Less)(R) as text "NegativeToRight(Less)(R)" end text end define end math ] "
\item " [ math define tex of lemma !!==Symmetry as text "!!==Symmetry" end text end define end math ] "
\item " [ math define tex of lemma negativeToRight(Eq)(R) as text "NegativeToRight(Eq)(R)" end text end define end math ] "
\item " [ math define tex of lemma negativeToRight(Eq)(1 term)(R) as text "NegativeToRight(Eq)(1 term)(R)" end text end define end math ] "
\item " [ math define tex of lemma doubleMinus(R) as text "DoubleMinus(R)" end text end define end math ] "
\item " [ math define tex of lemma uniqueNegative(R) as text "UniqueNegative(R)" end text end define end math ] "
\item " [ math define tex of lemma subtractEquationsLeft(R) as text "SubtractEquationsLeft(R)" end text end define end math ] "
\item " [ math define tex of lemma eqNegated(R) as text "EqNegated(R)" end text end define end math ] "
\item " [ math define tex of lemma neqNegated(R) as text "NeqNegated(R)" end text end define end math ] "
\item " [ math define tex of lemma subLeqRight(R) as text "SubLeqRight(R)" end text end define end math ] "
\item " [ math define tex of lemma leqNegated(R) as text "LeqNegated(R)" end text end define end math ] "
\item " [ math define tex of lemma lessNegated(R) as text "LessNegated(R)" end text end define end math ] "
\item " [ math define tex of lemma -0=0(R) as text "-0=0(R)" end text end define end math ] "
\item " [ math define tex of lemma negativeNegated(R) as text "NegativeNegated(R)" end text end define end math ] "
\item " [ math define tex of lemma from leqGeq(R) as text "FromLeqGeq(R)" end text end define end math ] "
\item " [ math define tex of lemma fromLess(R) as text "FromLess(R)" end text end define end math ] "
\item " [ math define tex of lemma nonnegativeNumerical(R) as text "NonnegativeNumerical(R)" end text end define end math ] "
\item " [ math define tex of lemma negativeNumerical(R) as text "NegativeNumerical(R)" end text end define end math ] "
\item " [ math define tex of lemma 0<=|x|(R) as text "0<=|x|(R)" end text end define end math ] "
\item " [ math define tex of lemma positiveNegated(R) as text "PositiveNegated(R)" end text end define end math ] "
\item " [ math define tex of lemma addEquations(R) as text "AddEquations(R)" end text end define end math ] "
\item " [ math define tex of lemma distributionOut(R) as text "DistributionOut(R)" end text end define end math ] "
\item " [ math define tex of lemma x*0+x=x(R) as text "x*0+x=x(R)" end text end define end math ] "

\item " [ math define tex of lemma x*0=0(R)fff as text "x*0=0(R)(fff)" end text end define end math ] "

\item " [ math define tex of lemma times(-1)(R) as text "Times(-1)(R)" end text end define end math ] "
\item " [ math define tex of lemma times(-1)Left(R) as text "Times(-1)Left(R)" end text end define end math ] "
\item " [ math define tex of lemma -x-y=-(x+y)(R) as text "-x-y=-(x+y)(R)" end text end define end math ] "
\item " [ math define tex of lemma lessTotality(R) as text "LessTotality(R)" end text end define end math ] "
\item " [ math define tex of lemma sameNumerical(R) as text "SameNumerical(R)" end text end define end math ] "
\item " [ math define tex of lemma minusNegated(R) as text "MinusNegated(R)" end text end define end math ] "
\item " [ math define tex of lemma positiveNumerical(R) as text "PositiveNumerical(R)" end text end define end math ] "
\item " [ math define tex of lemma signNumerical(+)(R) as text "SignNumerical(+)(R)" end text end define end math ] "
\item " [ math define tex of lemma signNumerical(R) as text "SignNumerical(R)" end text end define end math ] "
\item " [ math define tex of lemma numericalDifference(R) as text "NumericalDifference(R)" end text end define end math ] "
\item " [ math define tex of lemma x<=|x|(R) as text "x<=|x|(R)" end text end define end math ] "
\item " [ math define tex of lemma USlimitIsUpperBound helper as text "USlimitIsUpperBound(Helper)" end text end define end math ] "
\item " [ math define tex of lemma USlimitIsUpperBound as text "USlimitIsUpperBound" end text end define end math ] "
\item " [ math define tex of lemma (-1)*(-1)+(-1)*1=0(R) as text "(-1)*(-1)+(-1)*1=0(R)" end text end define end math ] "

\item " [ math define tex of lemma (-1)*(-1)=1(R) as text "(-1)*(-1)=1(R)" end text end define end math ] "

\item " [ math define tex of lemma 0<1Helper(R) as text "0<1Helper(R)" end text end define end math ] "
\item " [ math define tex of lemma 0<1(R) as text "0<1(R)" end text end define end math ] "
\item " [ math define tex of lemma expZero exact(R) as text "ExpZero(Exact)(R)" end text end define end math ] "
\item " [ math define tex of lemma positiveBase(R) base as text "PositiveBase(R)(Base)" end text end define end math ] "
\item " [ math define tex of lemma three2twoFactors(R) as text "Three2twoFactors(R)" end text end define end math ] "
\item " [ math define tex of lemma x=x*y*(1/y)(R) as text "x=x*y*(1/y)(R)" end text end define end math ] "

\item " [ math define tex of lemma neqMultiplication(R) as text "NeqMultiplication(R)" end text end define end math ] "
\item " [ math define tex of lemma lessTransitivity(R) as text "LessTransitivity(R)" end text end define end math ] "
\item " [ math define tex of lemma 0<2(R) as text "0<2(R)" end text end define end math ] "
\item " [ math define tex of lemma sameExp(R) base as text "SameExp(R)(Base)" end text end define end math ] "
\item " [ math define tex of lemma sameExp(R) indu as text "SameExp(R)(Indu)" end text end define end math ] "
\item " [ math define tex of lemma sameExp(R) as text "SameExp(R)" end text end define end math ] "
\item " [ math define tex of lemma subNeqLeft(R) as text "SubNeqLeft(R)" end text end define end math ] "
\item " [ math define tex of lemma subNeqRight(R) as text "SubNeqRight(R)" end text end define end math ] "
\item " [ math define tex of lemma nonzeroFactors(R) as text "NonzeroFactors(R)" end text end define end math ] "
\item " [ math define tex of lemma nonnegativeFactors(R) as text "NonnegativeFactors(R)" end text end define end math ] "
\item " [ math define tex of lemma positiveFactors(R) as text "PositiveFactors(R)" end text end define end math ] "
\item " [ math define tex of lemma lessDivision(R) as text "LessDivision(R)" end text end define end math ] "
\item " [ math define tex of lemma 0<1/2(R) as text "0<1/2(R)" end text end define end math ] "
\item " [ math define tex of lemma positiveToRight(Eq)(1 term)(R) as text "PositiveToRight(Eq)(1 term)(R)" end text end define end math ] "
\item " [ math define tex of lemma exp(+1)(R) as text "Exp(+1)(R)" end text end define end math ] "
\item " [ math define tex of lemma positiveBase(R) indu as text "PositiveBase(R)(Indu)" end text end define end math ] "
\item " [ math define tex of lemma positiveBase(R) as text "PositiveBase(R)" end text end define end math ] "
\item " [ math define tex of lemma -x*y=-(x*y)(R) as text "-x*y=-(x*y)(R)" end text end define end math ] "

\item " [ math define tex of lemma positiveToLeft(Eq)(R) as text "PositiveToLeft(Eq)(R)" end text end define end math ] "
\item " [ math define tex of lemma times1Left(R) as text "Times1Left(R)" end text end define end math ] "
\item " [ math define tex of lemma x+x=2*x(R) as text "x+x=2*x(R)" end text end define end math ] "

\item " [ math define tex of lemma (1/2)x+(1/2)x=x(R) as text "(1/2)x+(1/2)x=x(R)" end text end define end math ] "
\item " [ math define tex of lemma distributionOut(Minus)(R) as text "DistributionOut(Minus)(R)" end text end define end math ] "
\item " [ math define tex of lemma (1/2)(x+y)-x=(1/2)(y-x)(R) as text "(1/2)(x+y)-x=(1/2)(y-x)(R)" end text end define end math ] "
\item " [ math define tex of lemma intervalSize(R) base as text "IntervalSize(R)(Base)" end text end define end math ] "
\item " [ math define tex of lemma lessMultiplicationLeft(R) as text "LessMultiplicationLeft(R)" end text end define end math ] "
\item " [ math define tex of lemma negativeToLeft(Less)(R) as text "NegativeToLeft(Less)(R)" end text end define end math ] "
\item " [ math define tex of lemma negativeToLeft(Less)(1 term)(R) as text "NegativeToLeft(Less)(1 term)(R)" end text end define end math ] "
\item " [ math define tex of lemma y-(1/2)(x+y)=(1/2)(y-x)(R) as text "y-(1/2)(x+y)=(1/2)(y-x)(R)" end text end define end math ] "
\item " [ math define tex of lemma intervalSize(R) indu as text "IntervalSize(R)(Indu)" end text end define end math ] "
\item " [ math define tex of lemma intervalSize(R) as text "IntervalSize(R)" end text end define end math ] "
\item " [ math define tex of lemma XSlessUS(R) as text "XSlessUS(R)" end text end define end math ] "
\item " [ math define tex of lemma USdecreasing(+1)(R) as text "USdecreasing(+1)(R)" end text end define end math ] "
\item " [ math define tex of lemma 1<=x+1(N) as text "1<=x+1(N)" end text end define end math ] "
\item " [ math define tex of lemma expUnbounded base as text "ExpUnbounded(Base)" end text end define end math ] "
\item " [ math define tex of lemma expUnbounded indu as text "ExpUnbounded(Indu)" end text end define end math ] "
\item " [ math define tex of lemma expUnbounded as text "ExpUnbounded" end text end define end math ] "
\item " [ math define tex of lemma nonzeroProduct(2)(R) as text "NonzeroProduct(2)(R)" end text end define end math ] "
\item " [ math define tex of lemma positiveNonzero(R) as text "PositiveNonzero(R)" end text end define end math ] "
\item " [ math define tex of lemma nonreciprocalToRight(Eq)(1 term)(R) as text "NonreciprocalToRight(Eq)(1 term)(R)" end text end define end math ] "
\item " [ math define tex of lemma expNonzero base as text "ExpNonzero(Base)" end text end define end math ] "
\item " [ math define tex of lemma expNonzero indu as text "ExpNonzero(Indu)" end text end define end math ] "
\item " [ math define tex of lemma expNonzero as text "ExpNonzero" end text end define end math ] "
\item " [ math define tex of lemma multiplyEquations(R) as text "MultiplyEquations(R)" end text end define end math ] "
\item " [ math define tex of lemma expNonzero(2) as text "ExpNonzero(2)" end text end define end math ] "
\item " [ math define tex of lemma halfBase base as text "HalfBase(Base)" end text end define end math ] "
\item " [ math define tex of lemma halfBase indu as text "HalfBase(Indu)" end text end define end math ] "
\item " [ math define tex of lemma halfBase as text "HalfBase" end text end define end math ] "
\item " [ math define tex of lemma three2threeFactors(R) as text "Three2threeFactors(R)" end text end define end math ] "
\item " [ math define tex of lemma x*y=zBackwards(R) as text "x*y=zBackwards(R)" end text end define end math ] "

\item " [ math define tex of lemma positiveInverted(R) as text "PositiveInverted(R)" end text end define end math ] "
\item " [ math define tex of lemma reciprocalToRight(Less)(R) as text "ReciprocalToRight(Less)(R)" end text end define end math ] "
\item " [ math define tex of lemma reciprocalToRight(Less)(1 term)(R) as text "ReciprocalToRight(Less)(1 term)(R)" end text end define end math ] "
\item " [ math define tex of lemma nonreciprocalToLeft(Less)(R) as text "NonreciprocalToLeft(Less)(R)" end text end define end math ] "
\item " [ math define tex of lemma 1
\item " [ math define tex of lemma switchFactors(1/x \item " [ math define tex of lemma smallHalving as text "SmallHalving" end text end define end math ] "
\item " [ math define tex of lemma intervalSize(anyPositive) as text "IntervalSize(anyPositive)" end text end define end math ] "
\item " [ math define tex of lemma USdecreasing(+n) base as text "USdecreasing(+n)(Base)" end text end define end math ] "
\item " [ math define tex of lemma USdecreasing(+n) indu as text "USdecreasing(+n)(Indu)" end text end define end math ] "
\item " [ math define tex of lemma USdecreasing(+n) as text "USdecreasing(+n)" end text end define end math ] "
\item " [ math define tex of lemma USdecreasing as text "USdecreasing" end text end define end math ] "
\item " [ math define tex of lemma leqAdditionLeft(R) as text "LeqAdditionLeft(R)" end text end define end math ] "
\item " [ math define tex of lemma toNotLess(R) as text "ToNotLess(R)" end text end define end math ] "
\item " [ math define tex of lemma limitOfUSIsLeq as text "LimitOfUSIsLeq" end text end define end math ] "
\item " [ math define tex of lemma subtractEquations(Less)(R) as text "SubtractEquations(Less)(R)" end text end define end math ] "
\item " [ math define tex of lemma subtractEquationsLeft(Less)(R) as text "SubtractEquationsLeft(Less)(R)" end text end define end math ] "
\item " [ math define tex of lemma lessNegated(Negative)(R) as text "LessNegated(Negative)(R)" end text end define end math ] "
\item " [ math define tex of prop lemma from negated and (imply) as text "FromNegatedAnd(Imply)" end text end define end math ] "
\item " [ math define tex of prop lemma remove double neg (consequent) as text "RemoveDoubleNeg(Consequent)" end text end define end math ] "
\item " [ math define tex of lemma fromNotUpperBound as text "FromNotUpperBound" end text end define end math ] "
\item " [ math define tex of lemma leqNUB as text "LeqNUB" end text end define end math ] "
\item " [ math define tex of lemma USlimitIsLeastUpperBound helper as text "USlimitIsLeastUpperBound(Helper)" end text end define end math ] "
\item " [ math define tex of lemma USlimitIsLeastUpperBound as text "USlimitIsLeastUpperBound" end text end define end math ] "
\item " [ math define tex of pred lemma exist mp3 as text "ExistMP3" end text end define end math ] "
\item " [ math define tex of lemma greaterPositive(N) as text "GreaterPositive(N)" end text end define end math ] "
\item " [ math define tex of lemma ysFClose helper as text "ysFClose(Helper)" end text end define end math ] "
\item " [ math define tex of lemma ysFClose as text "ysFClose" end text end define end math ] "
\item " [ math define tex of lemma ysFCauchy helper as text "ysFCauchy(Helper)" end text end define end math ] "
\item " [ math define tex of lemma ysFCauchy as text "ysFCauchy" end text end define end math ] "

\item " [ math define tex of lemma from<<== as text "from<<==" end text end define end math ] "

\item " [ math define tex of lemma nonnegativeNumerical(F) as text "NonnegativeNumerical(F)" end text end define end math ] "

\item " [ math define tex of lemma to<<== as text "to<<==" end text end define end math ] "

\item " [ math define tex of lemma negativeNumerical(F) as text "NegativeNumerical(F)" end text end define end math ] "

\end{list}

\end{document}

End of file
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